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高等工程数学习题解答与提示

<link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/base.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/raw.css" rel="stylesheet"/><div id="sidebar" style="display: none"><div id="outline"></div></div><div class="pf w0 h0" data-page-no="1" id="pf1"><div class="pc pc1 w0 h0"><img alt="" class="bi x0 y0 w1 h1" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/bg1.jpg"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">高等工程数学</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws0">习题解答与提示</div><div class="t m0 x3 h4 y3 ff1 fs2 fc0 sc0 ls0 ws0">（教师内部参考）</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">教材<span class="ff2 fs3">——</span></div><div class="t m0 x4 h5 y5 ff1 fs4 fc0 sc0 ls0 ws0">南京理工大学高等工程数学编写组，高等工程数学讲义，<span class="ff3">2018</span>年<span class="ff3">7</span>月<span class="ff3">.</span></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div><div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/bg2.jpg"><div class="t m0 x5 h6 y6 ff1 fs5 fc0 sc0 ls0 ws0">目<span class="_ _0"></span>目<span class="_ _0"></span>目<span class="_ _1"> </span>录<span class="_ _0"></span>录<span class="_ _0"></span>录</div><div class="t m0 x4 h5 y7 ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>一<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _3"> </span><span class="ff4">1</span></div><div class="t m0 x4 h5 y8 ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>二<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _3"> </span><span class="ff4">9</span></div><div class="t m0 x4 h5 y9 ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>三<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">21</span></div><div class="t m0 x4 h5 ya ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>四<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">29</span></div><div class="t m0 x4 h5 yb ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>五<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">35</span></div><div class="t m0 x4 h5 yc ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>六<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">47</span></div><div class="t m0 x4 h5 yd ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>七<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">51</span></div><div class="t m0 x4 h5 ye ff1 fs4 fc0 sc0 ls0 ws0">第<span class="_ _2"> </span>八<span class="_ _2"> </span>章<span class="_"> </span>习题解答与提示<span class="_ _4"> </span><span class="ff4">55</span></div><div class="t m0 x6 h7 yf ff3 fs4 fc0 sc0 ls0 ws0">i</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div><div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/bg3.jpg"></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div><div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/bg4.jpg"><div class="t m0 x7 h5 y10 ff1 fs4 fc0 sc0 ls0 ws0">高等工程数学<span class="_ _5"> </span>第一章习题解答与提示<span class="_ _6"> </span><span class="ff3">1</span></div><div class="t m0 x8 h6 y11 ff1 fs5 fc0 sc0 ls0 ws0">第<span class="_ _0"></span>第<span class="_ _0"></span>第<span class="_ _7"> </span>一<span class="_ _0"></span>一<span class="_ _0"></span>一<span class="_ _7"> </span>章<span class="_ _0"></span>章<span class="_ _0"></span>章<span class="_ _8"> </span>习<span class="_ _0"></span>习<span class="_ _0"></span>习题<span class="_ _0"></span>题<span class="_ _0"></span>题解<span class="_ _0"></span>解<span class="_ _0"></span>解答<span class="_ _0"></span>答<span class="_ _0"></span>答与<span class="_ _0"></span>与<span class="_ _0"></span>与提<span class="_ _0"></span>提<span class="_ _0"></span>提示<span class="_ _9"></span>示<span class="_ _9"></span>示</div><div class="t m0 x9 h5 y12 ff3 fs4 fc0 sc0 ls0 ws0">1.<span class="_ _a"> </span><span class="ff1">分别证明例</span>1.1-<span class="ff1">例</span>1.4<span class="ff1">定义的距离都满足距离的三个条件。</span></div><div class="t m0 xa h5 y13 ff5 fs4 fc0 sc0 ls0 ws0">证明：<span class="ff1">例<span class="ff3">1.1</span>定义的离散距离满足距离都的三个条件是显然的，在此略去；</span></div><div class="t m0 xa h5 y14 ff1 fs4 fc0 sc0 ls0 ws0">例<span class="ff3">1.2</span>中<span class="ff6">d</span></div><div class="t m0 xb h8 y15 ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc h7 y14 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>d</div><div class="t m0 xd h8 y15 ff8 fs6 fc0 sc0 ls0 ws0">∞</div><div class="t m0 xe h5 y14 ff1 fs4 fc0 sc0 ls0 ws0">满足<span class="_ _c"></span>非负<span class="_ _c"></span>性、<span class="_ _c"></span>对称<span class="_ _c"></span>性由<span class="_ _c"></span>定义<span class="_ _c"></span>可以<span class="_ _c"></span>直接<span class="_ _c"></span>得到<span class="_ _c"></span>，由<span class="_ _c"></span>于绝<span class="_ _c"></span>对值<span class="_ _c"></span>满足<span class="_ _c"></span>三角</div><div class="t m0 xa h5 y16 ff1 fs4 fc0 sc0 ls0 ws0">不等式，<span class="_ _c"></span>所以<span class="ff6">d</span></div><div class="t m0 xf h8 y17 ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10 h7 y16 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>d</div><div class="t m0 x11 h8 y17 ff8 fs6 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x12 h5 y16 ff1 fs4 fc0 sc0 ls0 ws0">也满足三<span class="_ _c"></span>角不等式；<span class="_ _d"> </span>例<span class="ff3">1.2</span>中<span class="ff6">d</span></div><div class="t m0 x13 h8 y17 ff7 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x14 h5 y16 ff1 fs4 fc0 sc0 ls0 ws0">满足非负<span class="_ _c"></span>性、对称性<span class="_ _c"></span>由定义</div><div class="t m0 xa h5 y18 ff1 fs4 fc0 sc0 ls0 ws0">可以直<span class="_ _c"></span>接得到<span class="_ _c"></span>，取<span class="ff3">Minko<span class="_ _e"></span>wski<span class="ff1">不等式<span class="_ _c"></span>（见</span>1.6<span class="ff1">节）<span class="_ _c"></span>中<span class="ff6">p<span class="_ _f"> </span></span></span>=<span class="_ _f"> </span>2<span class="ff1">即可证<span class="_ _c"></span>明<span class="ff6">d</span></span></span></div><div class="t m0 x15 h8 y19 ff7 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x16 h5 y18 ff1 fs4 fc0 sc0 ls0 ws0">满足三<span class="_ _c"></span>角不</div><div class="t m0 xa h5 y1a ff1 fs4 fc0 sc0 ls0 ws0">等式<span class="ff3">;</span></div><div class="t m0 xa h5 y1b ff1 fs4 fc0 sc0 ls0 ws0">例<span class="ff3">1.3</span>中<span class="ff6">d</span></div><div class="t m0 xb h8 y1c ff9 fs6 fc0 sc0 ls0 ws0">p</div><div class="t m0 xc h5 y1b ff1 fs4 fc0 sc0 ls0 ws0">满<span class="_ _c"></span>足非<span class="_ _c"></span>负<span class="_ _c"></span>性<span class="_ _c"></span>、<span class="_ _c"></span>对称<span class="_ _c"></span>性<span class="_ _c"></span>由<span class="_ _c"></span>定<span class="_ _c"></span>义<span class="_ _c"></span>可以<span class="_ _c"></span>直<span class="_ _c"></span>接<span class="_ _c"></span>得<span class="_ _c"></span>到，<span class="_ _c"></span>利<span class="_ _c"></span>用<span class="ff3">Minko<span class="_ _e"></span>wski<span class="ff1">不<span class="_ _c"></span>等式<span class="_ _c"></span>即</span></span></div><div class="t m0 xa h5 y1d ff1 fs4 fc0 sc0 ls0 ws0">可证明<span class="ff6">d</span></div><div class="t m0 x17 h8 y1e ff9 fs6 fc0 sc0 ls0 ws0">p</div><div class="t m0 x18 h5 y1d ff1 fs4 fc0 sc0 ls0 ws0">满足三角不等式<span class="ff3">;</span></div><div class="t m0 xa h5 y1f ff1 fs4 fc0 sc0 ls0 ws0">例<span class="ff3">1.4</span>中<span class="ff6">d</span></div><div class="t m0 xb h8 y20 ff8 fs6 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x19 h5 y1f ff1 fs4 fc0 sc0 ls0 ws0">满<span class="_ _c"></span>足<span class="_ _c"></span>非<span class="_ _c"></span>负性<span class="_ _c"></span>、<span class="_ _c"></span>对<span class="_ _c"></span>称<span class="_ _c"></span>性<span class="_ _c"></span>由<span class="_ _c"></span>定<span class="_ _c"></span>义<span class="_ _c"></span>可<span class="_ _c"></span>以<span class="_ _c"></span>直<span class="_ _c"></span>接得<span class="_ _c"></span>到<span class="_ _c"></span>，<span class="_ _c"></span>利<span class="_ _c"></span>用<span class="_ _c"></span>上<span class="_ _c"></span>确<span class="_ _c"></span>界<span class="_ _c"></span>的<span class="_ _c"></span>定<span class="_ _c"></span>义<span class="_ _c"></span>以<span class="_ _c"></span>及</div><div class="t m0 xa h5 y21 ff1 fs4 fc0 sc0 ls0 ws0">绝对值的三角不等式，容易得到<span class="ff6">d</span></div><div class="t m0 x1a h8 y22 ff8 fs6 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x1b h5 y21 ff1 fs4 fc0 sc0 ls0 ws0">满足三角不等式。</div><div class="t m0 x9 h5 y23 ff3 fs4 fc0 sc0 ls0 ws0">2.<span class="_ _a"> </span><span class="ff1">证明极限的性质</span>1.1<span class="ff1">。</span></div><div class="t m0 xa h5 y24 ff5 fs4 fc0 sc0 ls0 ws0">证明：<span class="ff1">利用距离的三角不等式得：</span></div><div class="t m0 x1c h9 y25 ffa fs4 fc0 sc0 ls0 ws0">∥<span class="ff6">d<span class="ff3">(<span class="ffb">x</span></span></span></div><div class="t m0 x1d h8 y26 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x17 h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x1e h8 y27 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xd h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _10"> </span><span class="ffa">−<span class="_ _10"> </span><span class="ff6">d</span></span>(<span class="ffb">x</span></div><div class="t m0 x1f h8 y26 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x12 h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x20 h8 y27 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x21 h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="ffa">∥<span class="_ _b"></span>≤<span class="_ _b"></span>∥<span class="ff6">d</span></span>(<span class="ffb">x</span></div><div class="t m0 x22 h8 y26 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x23 h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x24 h8 y27 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x25 h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _10"> </span><span class="ffa">−<span class="_ _10"> </span><span class="ff6">d</span></span>(<span class="ffb">y</span></div><div class="t m0 x26 h8 y27 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x27 h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x28 h8 y26 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x29 h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="ffa">∥<span class="_ _10"> </span></span>+<span class="_ _10"> </span><span class="ffa">∥<span class="ff6">d</span></span>(<span class="ffb">y</span></div><div class="t m0 x2a h8 y27 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2b h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x2c h8 y26 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x2d h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _10"> </span><span class="ffa">−<span class="_ _10"> </span><span class="ff6">d</span></span>(<span class="ffb">x</span></div><div class="t m0 x2e h8 y26 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x2f h7 y25 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x30 h8 y27 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x31 h9 y25 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="ffa">∥</span></div><div class="t m0 x32 h9 y28 ffa fs4 fc0 sc0 ls0 ws0">≤<span class="_ _11"> </span><span class="ff6">d<span class="ff3">(<span class="ffb">x</span></span></span></div><div class="t m0 x33 h8 y29 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x34 h7 y28 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x35 h8 y29 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x36 h7 y28 ff3 fs4 fc0 sc0 ls0 ws0">))<span class="_ _10"> </span>+<span class="_ _10"> </span><span class="ff6">d</span>(<span class="ffb">y</span></div><div class="t m0 x27 h8 y2a ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x37 h7 y28 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x38 h8 y2a ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x39 h9 y28 ff3 fs4 fc0 sc0 ls0 ws0">))<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ff3">0<span class="ff6">,</span></span></span></div><div class="t m0 xa h5 y2b ff1 fs4 fc0 sc0 ls0 ws0">从而</div><div class="t m0 x3a h7 y2c ff3 fs4 fc0 sc0 ls0 ws0">lim</div><div class="t m0 x3b h8 y2d ff9 fs6 fc0 sc0 ls0 ws0">n<span class="ff8">→∞</span></div><div class="t m0 x3c h7 y2c ff6 fs4 fc0 sc0 ls0 ws0">d<span class="ff3">(<span class="ffb">x</span></span></div><div class="t m0 x3d h8 y2e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x3e h7 y2c ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x3f h8 y2f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x40 h7 y2c ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span>=<span class="_ _12"> </span><span class="ff6">d</span>(<span class="ffb">x</span></div><div class="t m0 x37 h8 y2e ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x41 h7 y2c ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x39 h8 y2f ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x14 h7 y2c ff3 fs4 fc0 sc0 ls0 ws0">)<span class="ff6">.</span></div><div class="t m0 xa h5 y30 ff1 fs4 fc0 sc0 ls0 ws0">若<span class="ffb">x</span></div><div class="t m0 x42 h8 y31 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x43 h7 y30 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x1d h8 y32 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xb h5 y30 ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _12"> </span><span class="ff6">X<span class="_ _14"></span><span class="ff1">都是点列</span></span>{<span class="ffb">x</span></div><div class="t m0 x44 h8 y31 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x45 h5 y30 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">的极限，则利用三角不等式得：</span></div><div class="t m0 x21 h7 y33 ff6 fs4 fc0 sc0 ls0 ws0">d<span class="ff3">(<span class="ffb">x</span></span></div><div class="t m0 x46 h8 y34 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x47 h7 y33 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x48 h8 y35 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x3c h9 y33 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span><span class="ffa">≤<span class="_ _12"> </span><span class="ff6">d</span></span>(<span class="ffb">x</span></div><div class="t m0 x49 h8 y34 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x4a h7 y33 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x4b h8 y34 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4c h7 y33 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _10"> </span>+<span class="_ _10"> </span><span class="ff6">d</span>(<span class="ffb">x</span></div><div class="t m0 x4d h8 y34 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4e h7 y33 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x4f h8 y35 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x50 h7 y33 ff3 fs4 fc0 sc0 ls0 ws0">)</div><div class="t m0 xa h5 y36 ff1 fs4 fc0 sc0 ls0 ws0">令<span class="ff6">n<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span>∞<span class="ff1">有<span class="ff6">d<span class="ff3">(<span class="ffb">x</span></span></span></span></span></span></div><div class="t m0 x51 h8 y37 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x52 h7 y36 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x53 h8 y37 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x54 h9 y36 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ff3">0<span class="ff6">,<span class="_ _b"> </span>d</span>(<span class="ffb">x</span></span></span></div><div class="t m0 x33 h8 y37 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x34 h7 y36 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x55 h8 y38 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x56 h5 y36 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ff3">0,<span class="ff1">因此<span class="_ _2"> </span><span class="ff6">d</span></span>(<span class="ffb">x</span></span></span></div><div class="t m0 x4f h8 y37 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x50 h7 y36 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x57 h8 y38 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x58 h5 y36 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span>=<span class="_ _12"> </span>0<span class="ff1">即<span class="ffb">x</span></span></div><div class="t m0 x59 h8 y37 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x5a h7 y36 ff3 fs4 fc0 sc0 ls0 ws0">=<span class="_ _12"> </span><span class="ffb">y</span></div><div class="t m0 x5b h8 y38 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x5c h5 y36 ff1 fs4 fc0 sc0 ls0 ws0">。</div><div class="t m0 x9 h5 y39 ff3 fs4 fc0 sc0 ls0 ws0">3.<span class="_ _a"> </span><span class="ff1">证明定理</span>1.1<span class="ff1">。</span></div><div class="t m0 xa h5 y3a ff5 fs4 fc0 sc0 ls0 ws0">证明：<span class="ff3">(1)<span class="_ _12"> </span>=<span class="_ _13"></span><span class="ffa">⇒<span class="_ _12"> </span><span class="ff3">(2)<span class="ff1">，由于<span class="ff6">T<span class="_ _15"> </span></span>是连续的，从而对</span></span>∀<span class="ff6">ε<span class="_ _12"> </span>><span class="_ _12"> </span><span class="ff3">0<span class="ff1">，存在</span></span>δ<span class="_ _f"> </span>><span class="_ _12"> </span><span class="ff3">0<span class="ff1">，当<span class="_ _10"> </span></span></span>d</span></span></span></div><div class="t m0 x5d h8 y3b ff9 fs6 fc0 sc0 ls0 ws0">X</div><div class="t m0 x2f h7 y3a ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">x<span class="ff6">,<span class="_ _b"> </span></span>x</span></div><div class="t m0 x5e h8 y3b ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x5f h7 y3a ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _12"> </span><span class="ff6"><</span></div><div class="t m0 xa h5 y3c ff6 fs4 fc0 sc0 ls0 ws0">δ<span class="_ _c"></span><span class="ff1">时<span class="_ _14"></span>，<span class="_ _14"></span>有</span>d</div><div class="t m0 xc h8 y3d ff9 fs6 fc0 sc0 ls0 ws0">Y</div><div class="t m0 x60 h7 y3c ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span>,<span class="_ _b"> </span>T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x2 h8 y3d ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x61 h5 y3c ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _16"> </span><span class="ff6"><<span class="_ _16"> </span>ε<span class="ff1">。<span class="_ _17"> </span>由<span class="_ _14"></span>于<span class="ffb">x</span></span></span></div><div class="t m0 x62 h8 y3d ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x36 h9 y3c ffa fs4 fc0 sc0 ls0 ws0">→<span class="_ _16"> </span><span class="ffb">x</span></div><div class="t m0 x63 h8 y3d ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x64 h5 y3c ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">n<span class="_ _16"> </span><span class="ffa">→<span class="_ _16"> </span>∞</span></span>)<span class="ff1">，<span class="_ _14"></span>对<span class="_ _14"></span>上<span class="_ _14"></span>述<span class="ff6">δ<span class="_ _a"> </span>><span class="_ _16"> </span></span></span>0<span class="ff1">存<span class="_ _14"></span>在<span class="_ _18"></span>自<span class="_ _14"></span>然</span></div><div class="t m0 xa h5 y3e ff1 fs4 fc0 sc0 ls0 ws0">数<span class="ff6">N<span class="_ _18"></span></span>，<span class="_ _c"></span>当<span class="ff6">n<span class="_ _f"> </span>><span class="_ _2"> </span>N<span class="_ _18"></span></span>时<span class="_ _d"> </span><span class="ff6">d</span></div><div class="t m0 x12 h8 y3f ff9 fs6 fc0 sc0 ls0 ws0">X</div><div class="t m0 x65 h7 y3e ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">x</span></div><div class="t m0 x66 h8 y3f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x44 h7 y3e ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x67 h8 y3f ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x68 h5 y3e ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _2"> </span><span class="ff6"><<span class="_ _f"> </span>δ<span class="_ _14"></span><span class="ff1">，所以</span>d</span></div><div class="t m0 x69 h8 y3f ff9 fs6 fc0 sc0 ls0 ws0">Y</div><div class="t m0 x4c h7 y3e ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x6a h8 y3f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6b h7 y3e ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>T<span class="_ _15"> </span><span class="ffb">x</span></div><div class="t m0 x6c h8 y3f ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x6d h5 y3e ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _2"> </span><span class="ff6"><<span class="_ _f"> </span>ε</span>,<span class="ff1">即<span class="ff6">T<span class="_ _b"> </span><span class="ffb">x</span></span></span></div><div class="t m0 x6e h8 y3f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6f h9 y3e ffa fs4 fc0 sc0 ls0 ws0">→<span class="_ _2"> </span><span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x70 h8 y3f ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x30 h9 y3e ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">n<span class="_ _2"> </span><span class="ffa">→</span></span></div><div class="t m0 xa h5 y40 ffa fs4 fc0 sc0 ls0 ws0">∞<span class="ff3">)<span class="ff1">。<span class="_ _2"> </span><span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></span></span></div><div class="t m0 x71 h8 y41 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x8 h9 y40 ffa fs4 fc0 sc0 ls0 ws0">→<span class="_ _12"> </span><span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x72 h8 y41 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x11 h5 y40 ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">n<span class="_ _12"> </span><span class="ffa">→<span class="_ _12"> </span>∞</span></span>)<span class="ff1">。</span></div><div class="t m0 xa h5 y42 ff3 fs4 fc0 sc0 ls0 ws0">(2)<span class="_ _17"> </span>=<span class="_ _13"></span><span class="ffa">⇒<span class="_ _19"> </span><span class="ff3">(1)<span class="ff1">，<span class="_ _14"></span>反<span class="_ _1a"></span>证<span class="_ _1a"></span>法<span class="_ _14"></span>。<span class="_ _1a"></span>若<span class="ff6">T<span class="_ _15"> </span></span>在<span class="ffb">x</span></span></span></span></div><div class="t m0 x73 h8 y43 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x74 h5 y42 ff1 fs4 fc0 sc0 ls0 ws0">不<span class="_ _1a"></span>连<span class="_ _14"></span>续<span class="_ _1a"></span>，<span class="_ _14"></span>则<span class="_ _1a"></span>存<span class="_ _14"></span>在<span class="ff6">ε</span></div><div class="t m0 x4d h8 y43 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x75 h5 y42 ff6 fs4 fc0 sc0 ls0 ws0">><span class="_ _19"> </span><span class="ff3">0<span class="ff1">，<span class="_ _14"></span>使<span class="_ _1a"></span>对<span class="_ _14"></span>任<span class="_ _1a"></span>意</span></span>δ<span class="_ _16"> </span>><span class="_ _17"> </span><span class="ff3">0<span class="ff1">，<span class="_ _1a"></span>存</span></span></div><div class="t m0 xa h5 y44 ff1 fs4 fc0 sc0 ls0 ws0">在<span class="ffb">x</span></div><div class="t m0 x42 h8 y45 ff9 fs6 fc0 sc0 ls0 ws0">δ</div><div class="t m0 x76 h5 y44 ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _16"> </span><span class="ff6">X<span class="_ _14"></span><span class="ff1">，<span class="_ _16"> </span>且</span>d</span></div><div class="t m0 x72 h8 y45 ff9 fs6 fc0 sc0 ls0 ws0">X</div><div class="t m0 x77 h7 y44 ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">x</span></div><div class="t m0 x54 h8 y45 ff9 fs6 fc0 sc0 ls0 ws0">δ</div><div class="t m0 x78 h7 y44 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x79 h8 y45 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x7a h5 y44 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _16"> </span><span class="ff6"><<span class="_ _a"> </span>δ<span class="_ _c"></span><span class="ff1">，<span class="_ _16"> </span>但</span>d</span></div><div class="t m0 x3f h8 y45 ff9 fs6 fc0 sc0 ls0 ws0">Y</div><div class="t m0 x7b h7 y44 ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x7c h8 y45 ff9 fs6 fc0 sc0 ls0 ws0">δ</div><div class="t m0 x7d h7 y44 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>T<span class="_ _15"> </span><span class="ffb">x</span></div><div class="t m0 x7e h8 y45 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x7f h9 y44 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _16"> </span><span class="ffa">≥<span class="_ _a"> </span><span class="ff6">ε</span></span></div><div class="t m0 x80 h8 y45 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x81 h5 y44 ff1 fs4 fc0 sc0 ls0 ws0">，<span class="_ _14"></span>特<span class="_ _14"></span>别<span class="_ _18"></span>取<span class="ff6">δ<span class="_ _1b"> </span><span class="ff3">=</span></span></div><div class="t m0 x82 h8 y46 ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x82 h8 y47 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x70 h7 y44 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ff3">(</span>n<span class="_ _16"> </span><span class="ff3">=</span></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div><div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89541243/bg5.jpg"><div class="t m0 x4 h5 y10 ff3 fs4 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span><span class="ff1">高等工程数学<span class="_ _5"> </span>第一章习题解答与提示</span></div><div class="t m0 xa h5 y48 ff3 fs4 fc0 sc0 ls0 ws0">1<span class="ff6">,<span class="_ _b"> </span></span>2<span class="ff6">,<span class="_ _b"> </span><span class="ffa">·<span class="_ _b"></span>·<span class="_ _b"></span>·<span class="_ _b"></span></span></span>)<span class="ff1">，<span class="_ _17"> </span>则<span class="_ _18"></span>有<span class="ffb">x</span></span></div><div class="t m0 x52 h8 y49 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x61 h5 y48 ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _16"> </span><span class="ff6">X<span class="_ _14"></span><span class="ff1">，</span>d</span></div><div class="t m0 x83 h8 y49 ff9 fs6 fc0 sc0 ls0 ws0">X</div><div class="t m0 x84 h7 y48 ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">x</span></div><div class="t m0 x85 h8 y49 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x86 h7 y48 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">x</span></div><div class="t m0 x87 h8 y49 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x88 h7 y48 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _16"> </span><span class="ff6"><</span></div><div class="t m0 x69 h8 y4a ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x69 h8 y4b ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4c h5 y48 ff1 fs4 fc0 sc0 ls0 ws0">，<span class="_ _17"> </span>但<span class="ff6">d</span></div><div class="t m0 x89 h8 y49 ff9 fs6 fc0 sc0 ls0 ws0">Y</div><div class="t m0 x14 h7 y48 ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x8a h8 y49 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x8b h7 y48 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>T<span class="_ _15"> </span><span class="ffb">x</span></div><div class="t m0 x8c h8 y49 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x8d h9 y48 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _16"> </span><span class="ffa">≥<span class="_ _a"> </span><span class="ff6">ε</span></span></div><div class="t m0 x16 h8 y49 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x8e h5 y48 ff1 fs4 fc0 sc0 ls0 ws0">，<span class="_ _17"> </span>这<span class="_ _14"></span>意<span class="_ _18"></span>味</div><div class="t m0 xa h5 y4c ff1 fs4 fc0 sc0 ls0 ws0">着<span class="ffb">x</span></div><div class="t m0 x42 h8 y4d ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x8f h9 y4c ffa fs4 fc0 sc0 ls0 ws0">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ffb">x</span></div><div class="t m0 x90 h8 y4d ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x1 h5 y4c ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">n<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span>∞<span class="ff3">)<span class="ff1">，但<span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></span></span></span></span></div><div class="t m0 x91 h8 y4d ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x85 h9 y4c ffa fs4 fc0 sc0 ls0 ws0">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ff6">T<span class="_ _15"> </span><span class="ffb">x</span></span></div><div class="t m0 x7b h8 y4d ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x92 h5 y4c ff3 fs4 fc0 sc0 ls0 ws0">(<span class="ff6">n<span class="_ _12"> </span><span class="ffa">−<span class="_ _13"></span>→<span class="_ _12"> </span>∞<span class="ff3">)<span class="ff1">不成立，矛盾。</span></span></span></span></div><div class="t m0 x9 h5 y4e ff3 fs4 fc0 sc0 ls0 ws0">4.<span class="_ _a"> </span><span class="ff1">证明例</span>1.16<span class="ff1">。</span></div><div class="t m0 xa h5 y4f ff5 fs4 fc0 sc0 ls0 ws0">证<span class="_ _1a"></span>明<span class="_ _14"></span>：<span class="ff3">(1).<span class="ff1">假<span class="_ _1a"></span>设<span class="ff6">X<span class="_ _14"></span></span>是<span class="_ _14"></span>有<span class="_ _1a"></span>限<span class="_ _14"></span>集<span class="_ _1a"></span>，<span class="ffa">{<span class="ffb">x</span></span></span></span></div><div class="t m0 x74 h8 y50 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x1a h5 y4f ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _19"> </span>⊂<span class="_ _17"> </span><span class="ff6">X<span class="_ _14"></span><span class="ff1">为<span class="_ _1a"></span>一<span class="_ _14"></span>无<span class="_ _1a"></span>穷<span class="_ _14"></span>点<span class="_ _1a"></span>列<span class="_ _14"></span>，<span class="_ _1a"></span>则</span>X<span class="_ _14"></span><span class="ff1">中<span class="_ _14"></span>至<span class="_ _1a"></span>少<span class="_ _14"></span>有<span class="_ _1a"></span>一<span class="_ _14"></span>元<span class="_ _1a"></span>素</span></span></div><div class="t m0 xa h5 y51 ff1 fs4 fc0 sc0 ls0 ws0">在<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 x93 h8 y52 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x94 h5 y51 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _f"> </span>⊂<span class="_ _f"> </span><span class="ff6">X<span class="_ _14"></span><span class="ff1">出<span class="_ _c"></span>现无<span class="_ _c"></span>穷多<span class="_ _c"></span>次，<span class="_ _c"></span>记该<span class="_ _c"></span>元素<span class="_ _c"></span>为<span class="ffb">x</span></span></span></div><div class="t m0 x4b h8 y52 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x95 h5 y51 ff1 fs4 fc0 sc0 ls0 ws0">。<span class="_ _2"> </span>从<span class="_ _c"></span>而在<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 x80 h8 y52 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x96 h5 y51 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">中可<span class="_ _c"></span>以选<span class="_ _c"></span>取子<span class="_ _c"></span>列，<span class="_ _c"></span>使</span></div><div class="t m0 xa h5 y53 ff1 fs4 fc0 sc0 ls0 ws0">得该子列每项都是<span class="ffb">x</span></div><div class="t m0 x12 h8 y54 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x97 h5 y53 ff1 fs4 fc0 sc0 ls0 ws0">，<span class="_ _2"> </span>显然该子列是收敛的。</div><div class="t m0 xa h5 y55 ff3 fs4 fc0 sc0 ls0 ws0">(2).<span class="ff1">不<span class="_ _1a"></span>妨<span class="_ _1a"></span>假<span class="_ _14"></span>设<span class="ffa">{<span class="ff6">x</span></span></span></div><div class="t m0 x98 h8 y56 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x99 h5 y55 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _19"> </span>⊂<span class="_ _19"> </span><span class="ff3">[<span class="ff6">a,<span class="_ _b"> </span>b</span>]<span class="ff1">为<span class="_ _14"></span>一<span class="_ _1a"></span>个<span class="_ _1a"></span>各<span class="_ _14"></span>元<span class="_ _1a"></span>素<span class="_ _1a"></span>互<span class="_ _14"></span>异<span class="_ _1a"></span>的<span class="_ _1a"></span>点<span class="_ _14"></span>列<span class="_ _1a"></span>，<span class="_ _1a"></span>对</span>[<span class="ff6">a,<span class="_ _b"> </span>b</span>]<span class="ff1">进<span class="_ _14"></span>行<span class="_ _1a"></span>二<span class="_ _1a"></span>等<span class="_ _14"></span>分<span class="_ _1a"></span>成<span class="_ _14"></span>两</span></span></div><div class="t m0 xa h5 y57 ff1 fs4 fc0 sc0 ls0 ws0">个<span class="_ _1a"></span>闭<span class="_ _14"></span>区<span class="_ _14"></span>间<span class="_ _1a"></span>，<span class="_ _14"></span>则<span class="_ _1a"></span>必<span class="_ _14"></span>有<span class="_ _1a"></span>一<span class="_ _14"></span>个<span class="_ _14"></span>区<span class="_ _1a"></span>间<span class="_ _14"></span>中<span class="_ _1a"></span>含<span class="_ _14"></span>有<span class="ffa">{<span class="ff6">x</span></span></div><div class="t m0 x40 h8 y58 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x9a h5 y57 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _19"> </span>⊂<span class="_ _17"> </span><span class="ff3">[<span class="ff6">a,<span class="_ _b"> </span>b</span>]<span class="ff1">中<span class="_ _14"></span>无<span class="_ _1a"></span>穷<span class="_ _14"></span>多<span class="_ _14"></span>个<span class="_ _1a"></span>元<span class="_ _14"></span>素<span class="_ _1a"></span>，<span class="_ _14"></span>记<span class="_ _1a"></span>该<span class="_ _14"></span>区<span class="_ _14"></span>间</span></span></div><div class="t m0 xa h5 y59 ff1 fs4 fc0 sc0 ls0 ws0">为<span class="ff3">[<span class="ff6">a</span></span></div><div class="t m0 x9b h8 y5a ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x93 h7 y59 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>b</div><div class="t m0 x1d h8 y5a ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x9c h5 y59 ff3 fs4 fc0 sc0 ls0 ws0">]<span class="ff1">。<span class="_ _c"></span>如<span class="_ _c"></span>此<span class="_ _c"></span>反<span class="_ _c"></span>复<span class="_ _c"></span>下<span class="_ _c"></span>去<span class="_ _c"></span>，<span class="_ _c"></span>我<span class="_ _c"></span>们<span class="_ _c"></span>得<span class="_ _c"></span>到<span class="_ _c"></span>一<span class="_ _c"></span>个<span class="_ _c"></span>闭<span class="_ _c"></span>区<span class="_ _c"></span>间<span class="_ _c"></span>序<span class="_ _c"></span>列<span class="ffa">{</span></span>[<span class="ff6">a</span></div><div class="t m0 x81 h8 y5a ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x9d h7 y59 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>b</div><div class="t m0 x2b h8 y5a ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x8c h5 y59 ff3 fs4 fc0 sc0 ls0 ws0">]<span class="ffa">}<span class="ff1">，<span class="_ _c"></span>该<span class="_ _c"></span>闭<span class="_ _c"></span>区<span class="_ _c"></span>间<span class="_ _c"></span>列<span class="_ _c"></span>中</span></span></div><div class="t m0 xa h5 y5b ff1 fs4 fc0 sc0 ls0 ws0">每一个闭区间都含有<span class="ffa">{<span class="ff6">x</span></span></div><div class="t m0 x21 h8 y5c ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x3 h5 y5b ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _12"> </span>⊂<span class="_ _12"> </span><span class="ff3">[<span class="ff6">a,<span class="_ _b"> </span>b</span>]<span class="ff1">中无穷多个元素，并且</span></span></div><div class="t m0 x1e h7 y5d ff3 fs4 fc0 sc0 ls0 ws0">[<span class="ff6">a</span></div><div class="t m0 x1 h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xe h7 y5d ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>b</div><div class="t m0 x9e h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x9f h9 y5d ff3 fs4 fc0 sc0 ls0 ws0">]<span class="_ _12"> </span><span class="ffa">⊃<span class="_ _12"> </span></span>[<span class="ff6">a</span></div><div class="t m0 x78 h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n<span class="ff7">+1</span></div><div class="t m0 xa0 h7 y5d ff6 fs4 fc0 sc0 ls0 ws0">b</div><div class="t m0 xa1 h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n<span class="ff7">+1</span></div><div class="t m0 x5 h9 y5d ff3 fs4 fc0 sc0 ls0 ws0">]<span class="ff6">,<span class="_ _b"> </span>n<span class="_ _12"> </span></span>=<span class="_ _12"> </span>1<span class="ff6">,<span class="_ _b"> </span></span>2<span class="ff6">,<span class="_ _b"> </span><span class="ffa">·<span class="_ _b"></span>·<span class="_ _b"></span>·<span class="_ _2"> </span></span>,<span class="_ _b"> </span>b</span></div><div class="t m0 xa2 h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x37 h9 y5d ffa fs4 fc0 sc0 ls0 ws0">−<span class="_ _10"> </span><span class="ff6">a</span></div><div class="t m0 x13 h8 y5e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa3 h9 y5d ffa fs4 fc0 sc0 ls0 ws0">−<span class="_ _13"></span>→<span class="_ _12"> </span><span class="ff3">0(<span class="ff6">n<span class="_ _12"> </span></span></span>−<span class="_ _13"></span>→<span class="_ _12"> </span>∞<span class="ff3">)<span class="ff6">.</span></span></div><div class="t m0 xa h5 y5f ff1 fs4 fc0 sc0 ls0 ws0">从而存在唯<span class="_ _c"></span>一一点<span class="ff6">x</span></div><div class="t m0 x12 h8 y60 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x54 h5 y5f ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _12"> </span><span class="ff3">[<span class="ff6">a,<span class="_ _b"> </span>b</span>]<span class="ff1">，使<span class="_ _c"></span>得对任何<span class="ff6">n</span>有<span class="ff6">x</span></span></span></div><div class="t m0 xa4 h8 y60 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x6b h9 y5f ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _12"> </span><span class="ff3">[<span class="ff6">a</span></span></div><div class="t m0 x6c h8 y60 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6d h7 y5f ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>b</div><div class="t m0 x80 h8 y60 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x81 h5 y5f ff3 fs4 fc0 sc0 ls0 ws0">](<span class="ff1">该性质称为<span class="_ _c"></span>闭区间套</span></div><div class="t m0 xa h5 y61 ff1 fs4 fc0 sc0 ls0 ws0">定理<span class="ff3">)</span>。根<span class="_ _c"></span>据构<span class="_ _c"></span>造，我<span class="_ _c"></span>们可以<span class="_ _c"></span>在<span class="ff3">[<span class="ff6">a</span></span></div><div class="t m0 x33 h8 y62 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x34 h7 y61 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span>b</div><div class="t m0 xa5 h8 y62 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa6 h5 y61 ff3 fs4 fc0 sc0 ls0 ws0">]<span class="ff1">中选取<span class="_ _c"></span><span class="ffa">{<span class="ff6">x</span></span></span></div><div class="t m0 xa7 h8 y62 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa8 h5 y61 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _2"> </span><span class="ff1">中<span class="_ _c"></span>互异的<span class="_ _c"></span>子列</span>{<span class="ff6">x</span></div><div class="t m0 x15 h8 y62 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa9 ha y63 ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 xaa h5 y61 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">，显然<span class="_ _c"></span>该</span></div><div class="t m0 xa h5 y64 ff1 fs4 fc0 sc0 ls0 ws0">点列以<span class="ff6">x</span></div><div class="t m0 x17 h8 y65 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x18 h5 y64 ff1 fs4 fc0 sc0 ls0 ws0">为极限。</div><div class="t m0 xa h5 y66 ff3 fs4 fc0 sc0 ls0 ws0">(3).<span class="_ _17"> </span><span class="ff1">略。</span></div><div class="t m0 xa h5 y67 ff3 fs4 fc0 sc0 ls0 ws0">(4).<span class="_ _17"> </span><span class="ff1">若<span class="_ _c"></span><span class="ff6">A<span class="_ _12"> </span><span class="ffa">⊂<span class="_ _12"> </span><span class="ffb">R</span></span></span></span></div><div class="t m0 xab h8 y68 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xac h5 y67 ff1 fs4 fc0 sc0 ls0 ws0">为有界闭集<span class="_ _c"></span>，按照<span class="ff3">(2)</span>的方法，<span class="_ _c"></span>可以同样证明<span class="ff6">A</span>为<span class="_ _c"></span>紧集。若<span class="ff6">A</span>为紧</div><div class="t m0 xa h5 y69 ff1 fs4 fc0 sc0 ls0 ws0">集<span class="_ _c"></span>，<span class="_ _c"></span>下<span class="_ _c"></span>证<span class="ff6">A</span>为<span class="_ _c"></span>有<span class="_ _c"></span>界<span class="_ _c"></span>闭<span class="_ _c"></span>集<span class="_ _c"></span>。<span class="_ _c"></span>首<span class="_ _c"></span>先<span class="_ _1a"></span>类<span class="_ _c"></span>似<span class="ff3">(3)</span>的<span class="_ _c"></span>方<span class="_ _c"></span>法<span class="_ _c"></span>，<span class="_ _c"></span>如<span class="_ _c"></span>果<span class="ff6">A</span>不<span class="_ _c"></span>是<span class="_ _c"></span>有<span class="_ _c"></span>界<span class="_ _c"></span>集<span class="_ _1a"></span>，<span class="_ _c"></span>可<span class="_ _c"></span>以<span class="_ _c"></span>构<span class="_ _c"></span>造<span class="ff6">A</span></div><div class="t m0 xa h5 y6a ff1 fs4 fc0 sc0 ls0 ws0">中一<span class="_ _c"></span>个点<span class="_ _c"></span>列<span class="_ _c"></span>，使<span class="_ _c"></span>之发<span class="_ _c"></span>散到<span class="_ _c"></span>无<span class="_ _c"></span>穷，<span class="_ _c"></span>该点<span class="_ _c"></span>列<span class="_ _c"></span>就不<span class="_ _c"></span>存在<span class="_ _c"></span>收敛<span class="_ _c"></span>子<span class="_ _c"></span>列。<span class="_ _c"></span>如果<span class="ff6">A</span>不<span class="_ _c"></span>是闭<span class="_ _c"></span>集<span class="_ _c"></span>，</div><div class="t m0 xa h5 y6b ff1 fs4 fc0 sc0 ls0 ws0">设<span class="ff6">x</span></div><div class="t m0 xad h8 y6c ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xae h5 y6b ff1 fs4 fc0 sc0 ls0 ws0">为<span class="ff6">A</span>的<span class="_ _c"></span>边<span class="_ _c"></span>界<span class="_ _c"></span>点<span class="_ _c"></span>，<span class="_ _1a"></span>则<span class="_ _c"></span>对<span class="_ _c"></span>任<span class="_ _c"></span>何<span class="_ _1a"></span>自<span class="_ _c"></span>然<span class="_ _c"></span>数<span class="ff6">n</span>，<span class="_ _1c"> </span><span class="ff6">B<span class="_ _1a"></span><span class="ff3">(</span>x</span></div><div class="t m0 x27 h8 y6c ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xaf h7 y6b ff6 fs4 fc0 sc0 ls0 ws0">,</div><div class="t m0 x41 h8 y6d ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x41 h8 y6e ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7e h9 y6b ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _1d"> </span><span class="ffa">∩<span class="_ _1d"> </span>{<span class="ff6">A<span class="_ _d"> </span></span>{<span class="ff6">x</span></span></div><div class="t m0 xb0 h8 y6c ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x2b h5 y6b ffa fs4 fc0 sc0 ls0 ws0">}}<span class="_ _d"> </span><span class="ff3">=<span class="_ _d"> </span></span>∅<span class="ff1">。<span class="_ _d"> </span>通<span class="_ _c"></span>过<span class="_ _1a"></span>适<span class="_ _c"></span>当</span></div><div class="t m0 xa h5 y6f ff1 fs4 fc0 sc0 ls0 ws0">选取，可以得到点列<span class="ffa">{<span class="ff6">x</span></span></div><div class="t m0 x21 h8 y70 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x3 h5 y6f ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">使得当<span class="ff6">n<span class="_ _12"> </span></span></span><span class="ff3">=<span class="_ _12"> </span><span class="ff6">m<span class="ff1">时</span>x</span></span></div><div class="t m0 xb1 h8 y70 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xb2 h9 y6f ffa fs4 fc0 sc0 ls0 ws0"><span class="ff3">=<span class="_ _12"> </span><span class="ff6">x</span></span></div><div class="t m0 x41 h8 y70 ff9 fs6 fc0 sc0 ls0 ws0">m</div><div class="t m0 xb3 h5 y6f ff1 fs4 fc0 sc0 ls0 ws0">且<span class="ff6">x</span></div><div class="t m0 xb4 h8 y70 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xb5 h9 y6f ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _12"> </span><span class="ff6">B<span class="_ _1a"></span><span class="ff3">(</span>x</span></div><div class="t m0 xb6 h8 y70 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xb7 h7 y6f ff6 fs4 fc0 sc0 ls0 ws0">,</div><div class="t m0 x2d h8 y71 ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2d h8 y72 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xb8 h9 y6f ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _1e"> </span><span class="ffa">∩<span class="_ _1e"></span>{<span class="ff6">A<span class="_ _f"> </span></span>{<span class="ff6">x</span></span></div><div class="t m0 x5c h8 y70 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xb9 h5 y6f ffa fs4 fc0 sc0 ls0 ws0">}}<span class="ff1">。</span></div><div class="t m0 xa h5 y73 ff1 fs4 fc0 sc0 ls0 ws0">根据构造可知<span class="ffa">{<span class="ff6">x</span></span></div><div class="t m0 xba h8 y74 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xbb h5 y73 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">在<span class="ff6">A</span>中不收敛，因为</span>{<span class="ff6">x</span></div><div class="t m0 xbc h8 y74 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xb1 h5 y73 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">收敛到<span class="ff6">A</span>的边界点<span class="ff6">x</span></span></div><div class="t m0 xbd h8 y74 ff7 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xbe h5 y73 ff1 fs4 fc0 sc0 ls0 ws0">。</div><div class="t m0 x9 h5 y75 ff3 fs4 fc0 sc0 ls0 ws0">5.<span class="_ _a"> </span><span class="ff1">证明定理</span>1.7<span class="ff1">。</span></div><div class="t m0 xa h5 y76 ff5 fs4 fc0 sc0 ls0 ws0">证明<span class="_ _c"></span>：<span class="ff3">(1)<span class="_ _d"> </span><span class="ff1">假<span class="_ _c"></span>设<span class="ff6">A</span>是<span class="_ _c"></span>距离<span class="_ _c"></span>空<span class="_ _c"></span>间<span class="ff6">X<span class="_ _14"></span></span>中<span class="_ _c"></span>的列<span class="_ _c"></span>紧<span class="_ _c"></span>集，<span class="ffa">{<span class="ffb">x</span></span></span></span></div><div class="t m0 x41 h8 y77 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7e h5 y76 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _2"> </span>∈<span class="_ _f"> </span><span class="ff6">A<span class="ff1">，<span class="_ _c"></span>且</span></span>{<span class="ffb">x</span></div><div class="t m0 xbf h8 y77 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2d h5 y76 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">各点<span class="_ _c"></span>互<span class="_ _c"></span>异<span class="_ _c"></span>。<span class="_ _d"> </span>如</span></div><div class="t m0 xa h5 y78 ff1 fs4 fc0 sc0 ls0 ws0">果<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 x93 h8 y79 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x94 h5 y78 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">中<span class="_ _c"></span>含<span class="_ _1a"></span>有<span class="_ _1a"></span>无<span class="_ _1a"></span>穷<span class="_ _1a"></span>多<span class="_ _1a"></span>个<span class="_ _1a"></span>点<span class="_ _1a"></span>在<span class="ff6">A</span>中<span class="_ _1a"></span>，<span class="_ _1a"></span>记</span>{<span class="ffb">x</span></div><div class="t m0 xc0 h8 y79 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc1 h9 y78 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _1d"> </span>∩<span class="_ _12"> </span><span class="ff6">A<span class="_ _1c"> </span><span class="ff3">=<span class="_ _1c"> </span></span></span>{<span class="ffb">x</span></div><div class="t m0 xc2 h8 y79 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc3 ha y7a ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 x8a h5 y78 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">，<span class="_ _c"></span>则<span class="_ _1a"></span>由<span class="_ _c"></span><span class="ff6">A</span>是<span class="_ _c"></span>列<span class="_ _1a"></span>紧<span class="_ _1a"></span>以<span class="_ _1a"></span>及<span class="ff6">A</span></span></div><div class="t m0 xa h5 y7b ff1 fs4 fc0 sc0 ls0 ws0">包含<span class="ff6">A</span>的<span class="_ _c"></span>所<span class="_ _c"></span>有聚<span class="_ _c"></span>点<span class="_ _c"></span>，可<span class="_ _c"></span>以<span class="_ _c"></span>得到<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 xc4 h8 y7c ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x1a ha y7d ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 xc5 h5 y7b ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">在<span class="ff6">A</span>中有<span class="_ _c"></span>收<span class="_ _c"></span>敛子<span class="_ _c"></span>列<span class="_ _c"></span>，从<span class="_ _c"></span>而</span>{<span class="ffb">x</span></div><div class="t m0 xbe h8 y7c ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc6 h5 y7b ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">在<span class="ff6">A</span>中有<span class="_ _c"></span>收<span class="_ _c"></span>敛</span></div><div class="t m0 xa h5 y7e ff1 fs4 fc0 sc0 ls0 ws0">子<span class="_ _1a"></span>列<span class="_ _1a"></span>。<span class="_ _19"> </span>如<span class="_ _c"></span>果<span class="_ _c"></span><span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 x9e h8 y7f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x9f h5 y7e ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">中<span class="_ _1a"></span>不<span class="_ _1a"></span>含<span class="_ _1a"></span>有<span class="_ _1a"></span>无<span class="_ _1a"></span>穷<span class="_ _1a"></span>多<span class="_ _14"></span>个<span class="_ _c"></span>点<span class="_ _14"></span>在<span class="ff6">A</span>中<span class="_ _c"></span>，<span class="_ _1a"></span>则</span>{<span class="_"> </span><span class="ffb">x</span></div><div class="t m0 xc2 h8 y7f ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xb5 h5 y7e ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">中<span class="_ _1a"></span>含<span class="_ _1a"></span>有<span class="_ _1a"></span>无<span class="_ _1a"></span>穷<span class="_ _1a"></span>多<span class="_ _1a"></span>个<span class="ff6">A</span>的<span class="_ _14"></span>边</span></div><div class="t m0 xa h5 y80 ff1 fs4 fc0 sc0 ls0 ws0">界<span class="_ _c"></span>点<span class="_ _c"></span>，<span class="_ _c"></span>记<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 xc7 h8 y81 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc8 h5 y80 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">中<span class="_ _c"></span>所<span class="_ _c"></span>有<span class="_ _c"></span>边<span class="_ _c"></span>界<span class="_ _c"></span>点<span class="_ _c"></span>组<span class="_ _c"></span>成<span class="_ _c"></span>的<span class="_ _c"></span>子<span class="_ _c"></span>列<span class="_ _c"></span>为<span class="_ _1c"> </span></span>{<span class="ffb">x</span></div><div class="t m0 xc9 h8 y81 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6a ha y82 ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 xa8 h5 y80 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">，<span class="_ _c"></span>对<span class="_ _c"></span>每<span class="_ _c"></span>个<span class="ffb">x</span></span></div><div class="t m0 xca h8 y81 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x8d ha y82 ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 xbd h5 y80 ff1 fs4 fc0 sc0 ls0 ws0">，<span class="_ _c"></span>存<span class="_ _c"></span>在<span class="ffb">y</span></div><div class="t m0 xcb h8 y83 ff9 fs6 fc0 sc0 ls0 ws0">k</div><div class="t m0 x5b h5 y80 ffa fs4 fc0 sc0 ls0 ws0">∈<span class="_ _2"> </span><span class="ff6">A<span class="ff1">使</span></span></div><div class="t m0 xa h5 y84 ff1 fs4 fc0 sc0 ls0 ws0">得<span class="ff6">d<span class="ff3">(<span class="ffb">x</span></span></span></div><div class="t m0 x76 h8 y85 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xcc ha y86 ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 x9c h7 y84 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _b"> </span><span class="ffb">y</span></div><div class="t m0 x19 h8 y87 ff9 fs6 fc0 sc0 ls0 ws0">k</div><div class="t m0 xcd h7 y84 ff3 fs4 fc0 sc0 ls0 ws0">)<span class="_ _1b"> </span><span class="ff6"><</span></div><div class="t m0 x99 h8 y88 ff7 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x99 h8 y89 ff9 fs6 fc0 sc0 ls0 ws0">k</div><div class="t m0 xce h5 y84 ff1 fs4 fc0 sc0 ls0 ws0">。<span class="_ _18"> </span>由<span class="_ _15"> </span>于<span class="ff6">A</span>是<span class="_ _15"> </span>列<span class="_ _18"></span>紧<span class="_ _15"> </span>以<span class="_ _18"></span>及<span class="ff6">A<span class="_ _a"> </span></span>包<span class="_ _18"> </span>含<span class="ff6">A</span>的<span class="_ _15"> </span>所<span class="_ _18"> </span>有<span class="_ _15"> </span>聚<span class="_ _15"> </span>点<span class="_ _18"></span>，<span class="_ _15"> </span>所<span class="_ _18"></span>以<span class="ffa">{<span class="ffb">y</span></span></div><div class="t m0 xcf h8 y87 ff9 fs6 fc0 sc0 ls0 ws0">k</div><div class="t m0 xd0 h5 y84 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">在<span class="_ _18"> </span>存</span></div><div class="t m0 xa h5 y8a ff1 fs4 fc0 sc0 ls0 ws0">在<span class="ff6">A</span>中的收敛子列<span class="ffa">{<span class="ffb">y</span></span></div><div class="t m0 x53 h8 y8b ff9 fs6 fc0 sc0 ls0 ws0">k</div><div class="t m0 x54 ha y8c ffc fs7 fc0 sc0 ls0 ws0">m</div><div class="t m0 xd1 h5 y8a ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">，根据构造可以得到<span class="ffb">x</span></span></div><div class="t m0 xd2 h8 y8d ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xa7 ha y8e ffc fs7 fc0 sc0 ls0 ws0">k</div><div class="t m0 xd3 ha y8f ffc fs7 fc0 sc0 ls0 ws0">m</div><div class="t m0 xd4 h5 y8a ff1 fs4 fc0 sc0 ls0 ws0">在<span class="ff6">A</span>中收敛。<span class="_ _f"> </span>所以<span class="ff6">A</span>为紧集。</div><div class="t m0 xa h5 y90 ff3 fs4 fc0 sc0 ls0 ws0">(2)<span class="_ _2"> </span><span class="ff1">假设<span class="ff6">A</span>是距离空<span class="_ _c"></span>间<span class="ff6">X<span class="_ _14"></span></span>中的列紧集，<span class="ff6">B<span class="_ _d"> </span><span class="ffa">⊂<span class="_ _12"> </span></span>A</span>。若<span class="ffa">{<span class="ffb">x</span></span></span></div><div class="t m0 x4e h8 y91 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6c h5 y90 ffa fs4 fc0 sc0 ls0 ws0">}<span class="_ _12"> </span>⊂<span class="_ _12"> </span><span class="ff6">B<span class="_ _d"> </span></span>⊂<span class="_ _12"> </span><span class="ff6">A<span class="ff1">，则由于</span>A<span class="ff1">是<span class="_ _c"></span>列</span></span></div><div class="t m0 xa h5 y92 ff1 fs4 fc0 sc0 ls0 ws0">紧的，从而<span class="ffa">{<span class="ffb">x</span></span></div><div class="t m0 xd5 h8 y93 ff9 fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 xd6 h5 y92 ffa fs4 fc0 sc0 ls0 ws0">}<span class="ff1">有子列<span class="_ _2"> </span>收敛到<span class="ff6">X<span class="_ _14"></span></span>中，根据定义可知<span class="ff6">B<span class="_ _1a"></span></span>为列紧集。</span></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>

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