Cauchy序列GydF4y2Ba/h1>
已经有一个帐户?一种href="//www.parkandroid.com/account/login/?next=/wiki/cauchy-sequences/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">在此登录。GydF4y2Ba/a>
测验GydF4y2Ba/h4>
与...相关GydF4y2Ba/h4>
- 几何学GydF4y2Ba/span>>GydF4y2Ba/span>
一种GydF4y2Bastrong>Cauchy序列GydF4y2Ba/strong>是A.一种href="//www.parkandroid.com/wiki/terminology-of-sequences-and-series/" class="wiki_link" title="顺序GydF4y2Ba" target="_blank">顺序GydF4y2Ba/a>随着序列的进展,其术语彼此非常接近。正式,序列S.P.一种N.class="katex">
{GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>}GydF4y2Ba/span>N.GydF4y2Ba/span>=GydF4y2Ba/span>0.GydF4y2Ba/span>∞GydF4y2Ba/span>是一个cauchy序列,如果为每一个GydF4y2Baspan class="katex">
ε.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>那GydF4y2Ba/span>有一个GydF4y2Baspan class="katex">
N.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>这样GydF4y2Baspan class="katex-display">
N.GydF4y2Ba/span>那GydF4y2Ba/span>M.GydF4y2Ba/span>>GydF4y2Ba/span>N.GydF4y2Ba/span>⟹GydF4y2Ba/span>|GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>|GydF4y2Ba/span>GydF4y2Ba/span>ε.GydF4y2Ba/span>。GydF4y2Ba/span>翻译符号,这意味着对于任何小距离,存在某种指标过去,任何两个术语在该距离中的距离内部,这捕获了术语的直观思想。这也可以写成S.P.一种N.class="katex-display">
M.GydF4y2Ba/span>那GydF4y2Ba/span>N.GydF4y2Ba/span>L.GydF4y2Ba/span>一世GydF4y2Ba/span>M.GydF4y2Ba/span>S.GydF4y2Ba/span>你GydF4y2Ba/span>P.GydF4y2Ba/span>|GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>=GydF4y2Ba/span>0.GydF4y2Ba/span>那GydF4y2Ba/span>在那里一种href="//www.parkandroid.com/wiki/infimium/" class="wiki_link" title="限制上级GydF4y2Ba" target="_blank">限制上级GydF4y2Ba/a>正在采取。GydF4y2Ba/p>
Cauchy序列是有用的,因为它们引起了一个概念一种href="//www.parkandroid.com/wiki/metric-space/" class="wiki_link" title="完全的GydF4y2Ba" target="_blank">完全的GydF4y2Ba/a>场地GydF4y2Ba/a>,这是每个Cauchy序列会聚的领域。因为Cauchy序列是序列,其术语在一起靠近,所以所有Cauchy序列收敛的字段是不是“缺少”任何数字的字段。规范完整的领域是S.P.一种N.class="katex">
R.GydF4y2Ba/span>,因此了解Cauchy序列对于了解性质和结构是必不可少的GydF4y2Baspan class="katex">
R.GydF4y2Ba/span>。GydF4y2Ba/p>
内容GydF4y2Ba/h4>
Cauchy序列在GydF4y2Baspan class="katex"> R.GydF4y2Ba/span>
上面给出的Cauchy序列的定义可用于将序列鉴定为Cauchy序列。GydF4y2Ba/p>
是序列GydF4y2Baspan class="katex"> 一种GydF4y2Ba/span>N.GydF4y2Ba/span>=GydF4y2Ba/span>2GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>一个cauchy序列?GydF4y2Ba/p>
是的。采取任何S.P.一种N.class="katex"> ε.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>,选择GydF4y2Baspan class="katex"> N.GydF4y2Ba/span>太大了GydF4y2Baspan class="katex"> 2GydF4y2Ba/span>-GydF4y2Ba/span>N.GydF4y2Ba/span>GydF4y2Ba/span>ε.GydF4y2Ba/span>。然后,如果S.P.一种N.class="katex"> N.GydF4y2Ba/span>那GydF4y2Ba/span>M.GydF4y2Ba/span>>GydF4y2Ba/span>N.GydF4y2Ba/span>, 我们有GydF4y2Baspan class="katex-display"> |GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>|GydF4y2Ba/span>=GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>2GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>-GydF4y2Ba/span>2GydF4y2Ba/span>M.GydF4y2Ba/span>1GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>|GydF4y2Ba/span>≤.GydF4y2Ba/span>2GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>+GydF4y2Ba/span>2GydF4y2Ba/span>M.GydF4y2Ba/span>1GydF4y2Ba/span>≤.GydF4y2Ba/span>2GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>+GydF4y2Ba/span>2GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>=GydF4y2Ba/span>ε.GydF4y2Ba/span>那GydF4y2Ba/span>所以这个序列是Cauchy。GydF4y2Baspan class="katex"> □GydF4y2Ba/span>
显示序列是GydF4y2Baem>不是GydF4y2Ba/em>Cauchy有点棘手。对于不成为Cauchy的序列,需要一些S.P.一种N.class="katex">
N.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>这样的GydF4y2Baspan class="katex">
ε.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>, 有GydF4y2Baspan class="katex">
M.GydF4y2Ba/span>那GydF4y2Ba/span>N.GydF4y2Ba/span>>GydF4y2Ba/span>N.GydF4y2Ba/span>和GydF4y2Baspan class="katex">
|GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>|GydF4y2Ba/span>>GydF4y2Ba/span>ε.GydF4y2Ba/span>。换句话说,无论序列有多远,术语都没有保证,他们将靠近在一起。GydF4y2Ba/p>
是序列GydF4y2Baspan class="katex">
一种GydF4y2Ba/span>N.GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>一个cauchy序列?GydF4y2Ba/p>
不GydF4y2Baspan class="katex">
ε.GydF4y2Ba/span>=GydF4y2Ba/span>1GydF4y2Ba/span>。然后,任何S.P.一种N.class="katex">
N.GydF4y2Ba/span>,如果我们采取GydF4y2Baspan class="katex">
N.GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>+GydF4y2Ba/span>3.GydF4y2Ba/span>和GydF4y2Baspan class="katex">
M.GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>+GydF4y2Ba/span>1GydF4y2Ba/span>,我们有那个GydF4y2Baspan class="katex">
|GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>=GydF4y2Ba/span>2GydF4y2Ba/span>>GydF4y2Ba/span>1GydF4y2Ba/span>,所以从来没有任何东西GydF4y2Baspan class="katex">
N.GydF4y2Ba/span>这适用于此GydF4y2Baspan class="katex">
ε.GydF4y2Ba/span>。GydF4y2Ba/span>因此,序列不是Cauchy。GydF4y2Baspan class="katex">
□GydF4y2Ba/span>
在尝试确定序列是否是Cauchy时,最容易使用越来越多地靠近的术语的直觉来决定它是否是,然后使用该定义来证明它。GydF4y2Ba/p>
在抽象公制空间中的cauchy序列GydF4y2Ba/h2>
前一段的思想可用于考虑一般的Cauchy序列一种href="//www.parkandroid.com/wiki/metric-space/" class="wiki_link" title="公制空间GydF4y2Ba" target="_blank">公制空间GydF4y2Ba/a>
(GydF4y2Ba/span>XGydF4y2Ba/span>那GydF4y2Ba/span>D.GydF4y2Ba/span>的)GydF4y2Ba/span>。GydF4y2Ba/span>在这种情况下,一个序列GydF4y2Baspan class="katex">
{GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>}GydF4y2Ba/span>据说是Cauchy,如果为每一个GydF4y2Baspan class="katex">
ε.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>, 那里存在GydF4y2Baspan class="katex">
N.GydF4y2Ba/span>>GydF4y2Ba/span>0.GydF4y2Ba/span>这样GydF4y2Baspan class="katex-display">
M.GydF4y2Ba/span>那GydF4y2Ba/span>N.GydF4y2Ba/span>>GydF4y2Ba/span>N.GydF4y2Ba/span>⟹GydF4y2Ba/span>D.GydF4y2Ba/span>(GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>那GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>的)GydF4y2Ba/span>GydF4y2Ba/span>ε.GydF4y2Ba/span>。GydF4y2Ba/span>除了直观的水平,除了使用“距离”的概念之外,没有任何改变。GydF4y2Ba/p>
考虑连续功能的公制空间GydF4y2Baspan class="katex">
[GydF4y2Ba/span>0.GydF4y2Ba/span>那GydF4y2Ba/span>1GydF4y2Ba/span>]GydF4y2Ba/span>用公制GydF4y2Baspan class="katex-display">
D.GydF4y2Ba/span>(GydF4y2Ba/span>FGydF4y2Ba/span>那GydF4y2Ba/span>GGydF4y2Ba/span>的)GydF4y2Ba/span>=GydF4y2Ba/span>∫GydF4y2Ba/span>0.GydF4y2Ba/span>1GydF4y2Ba/span>|GydF4y2Ba/span>FGydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>-GydF4y2Ba/span>GGydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>|GydF4y2Ba/span>D.GydF4y2Ba/span>XGydF4y2Ba/span>。GydF4y2Ba/span>是序列GydF4y2Baspan class="katex">
FGydF4y2Ba/span>N.GydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>XGydF4y2Ba/span>这个空间的Cauchy序列?GydF4y2Ba/p>
注意GydF4y2Baspan class="katex-display">
D.GydF4y2Ba/span>(GydF4y2Ba/span>FGydF4y2Ba/span>M.GydF4y2Ba/span>那GydF4y2Ba/span>FGydF4y2Ba/span>N.GydF4y2Ba/span>的)GydF4y2Ba/span>=GydF4y2Ba/span>∫GydF4y2Ba/span>0.GydF4y2Ba/span>1GydF4y2Ba/span>|GydF4y2Ba/span>M.GydF4y2Ba/span>XGydF4y2Ba/span>-GydF4y2Ba/span>N.GydF4y2Ba/span>XGydF4y2Ba/span>|GydF4y2Ba/span>D.GydF4y2Ba/span>XGydF4y2Ba/span>=GydF4y2Ba/span>[GydF4y2Ba/span>|GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>2GydF4y2Ba/span>XGydF4y2Ba/span>2GydF4y2Ba/span>]GydF4y2Ba/span>0.GydF4y2Ba/span>1GydF4y2Ba/span>=GydF4y2Ba/span>2GydF4y2Ba/span>|GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>。GydF4y2Ba/span>通过服用GydF4y2Baspan class="katex">
M.GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>+GydF4y2Ba/span>1GydF4y2Ba/span>,我们总是可以做到这一点GydF4y2Baspan class="katex">
2GydF4y2Ba/span>1GydF4y2Ba/span>,所以至少有术语至少GydF4y2Baspan class="katex">
2GydF4y2Ba/span>1GydF4y2Ba/span>分开,因此这个序列不是Cauchy。GydF4y2Baspan class="katex">
□GydF4y2Ba/span>
考虑由连续功能组成的度量空间GydF4y2Baspan class="katex">
[GydF4y2Ba/span>0.GydF4y2Ba/span>那GydF4y2Ba/span>1GydF4y2Ba/span>]GydF4y2Ba/span>用公制GydF4y2Baspan class="katex-display">
D.GydF4y2Ba/span>(GydF4y2Ba/span>FGydF4y2Ba/span>那GydF4y2Ba/span>GGydF4y2Ba/span>的)GydF4y2Ba/span>=GydF4y2Ba/span>∫GydF4y2Ba/span>0.GydF4y2Ba/span>1GydF4y2Ba/span>|GydF4y2Ba/span>FGydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>-GydF4y2Ba/span>GGydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>|GydF4y2Ba/span>D.GydF4y2Ba/span>XGydF4y2Ba/span>。GydF4y2Ba/span>是序列GydF4y2Baspan class="katex">
FGydF4y2Ba/span>N.GydF4y2Ba/span>(GydF4y2Ba/span>XGydF4y2Ba/span>的)GydF4y2Ba/span>=GydF4y2Ba/span>N.GydF4y2Ba/span>XGydF4y2Ba/span>这个空间的Cauchy序列?GydF4y2Ba/p>
Cauchy序列和收敛GydF4y2Ba/h2>
Cauchy序列与收敛序列密切相关。例如,每个会聚序列都是Cauchy,因为如果S.P.一种N.class="katex">
一种GydF4y2Ba/span>N.GydF4y2Ba/span>→GydF4y2Ba/span>XGydF4y2Ba/span>, 然后GydF4y2Baspan class="katex-display">
|GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>≤.GydF4y2Ba/span>|GydF4y2Ba/span>一种GydF4y2Ba/span>M.GydF4y2Ba/span>-GydF4y2Ba/span>XGydF4y2Ba/span>|GydF4y2Ba/span>+GydF4y2Ba/span>|GydF4y2Ba/span>XGydF4y2Ba/span>-GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>|GydF4y2Ba/span>那GydF4y2Ba/span>这两个都必须转到零。这个问题的悔改,无论是每个Cauchy序列是否收敛,都会导致以下定义:GydF4y2Ba/p>
如果字段中的每个Cauchy序列会聚到字段的元素,则完成一个字段。GydF4y2Ba/p>
是GydF4y2Baspan class="katex">
问:GydF4y2Ba/span>一个完整的领域?GydF4y2Ba/p>
不。通过序列给出GydF4y2Baspan class="katex">
一种GydF4y2Ba/span>0.GydF4y2Ba/span>=GydF4y2Ba/span>1GydF4y2Ba/span>并满意GydF4y2Baspan class="katex">
一种GydF4y2Ba/span>N.GydF4y2Ba/span>=GydF4y2Ba/span>2GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>-GydF4y2Ba/span>1GydF4y2Ba/span>+GydF4y2Ba/span>一种GydF4y2Ba/span>N.GydF4y2Ba/span>1GydF4y2Ba/span>。这个序列有极限S.P.一种N.class="katex">
2GydF4y2Ba/span>
,所以它是cauchy,但这个限制不在GydF4y2Baspan class="katex">
问:GydF4y2Ba/span>那GydF4y2Ba/span>所以GydF4y2Baspan class="katex">
问:GydF4y2Ba/span>不是一个完整的领域。GydF4y2Baspan class="katex">
□GydF4y2Ba/span>
尤其,GydF4y2Baspan class="katex">
R.GydF4y2Ba/span>是一个完整的领域,这个事实形成了大部分实际分析的基础:要显示一系列实数会收敛,只需要表明它是Cauchy。同样,鉴于Cauchy序列,它会自动限制,这是广泛适用的事实。GydF4y2Ba/p>
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注意加载......GydF4y2Ba/p>
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