---- > [!definition] Definition. ([[Cauchy sequence]]) > A [[sequence]] $x_{1},x_{2},\dots$ in a [[metric space]] $(X,d)$ is called **cauchy** if for all $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that for all $n_{1},n_{2} \geq N$ one has $d(x_{n_{1}}, x_{n_{2}})<\varepsilon.$ ^definition > [!basicproperties] > - [[Cauchy sequence is convergent iff has convergent subsequence]] > > > - (Thinning until summable) For $X$ induced by a [[norm]], if $(x_{n})$ is a Cauchy sequence, then $(x_{n})$ has a subsequence $(x_{n_{k}})$ satisfying $\sum_{k=1}^{\infty} \|x_{n_{k}}-x_{n_{k-1}}\|<\infty.$ > > > > > [!proof]- > > Indeed, since $(x_{n})$ is Cauchy, for each $j \in \mathbb{N}$ there exists $N_{j}$ such that $\|x_{n}-x_{m}\|<2^{-j}$ for all $m ,n \geq N_{j}$. Now define $x_{n_{k}}$ as follows: > > - Choose $n_{1} \geq N_{1}$. > > - Having chosen $n_{1}$, choose $n_{2} > n_{1}$ with $n_{2} \geq N_{2}$ > > - and so on: choose $n_{k} > \dots > n_{2}>n_{1}$ with $n_{k} \geq N_{k}$. > > > > Then $\|x_{n_{k}}-x_{n_{k-1}}\|<2^{-(k-1)}$. Hence $\sum_{k=1}^{\infty}\|x_{n_{k}}-x_{n_{k-1}}\|$ is bounded by a standard geometric series, hence converges. > > > >- If the metric on $X$ is induced by a [[norm]], then every [[Cauchy sequence]] in $X$ is bounded > > > [!proof]- Proof. > > > > Suppose $X$ is a [[normed vector space]] and $(x_{n})$ is a [[sequence]] in $X$. Obtain $N \in \mathbb{N}$ such that for all $n,m \geq N$ one has $\|x_{n}-x_{m}\| \leq \frac{1}{59}$. In particular, $\|x_{n}-x_{N}\| \leq \frac{1}{59}$ for all $n \geq N$. It follows that $\|x_{n}\| \leq \|x_{N}\|+\frac{1}{59}$ for all $n \geq N$. Now the value $b:=\max \left\{ \|x_{1}\|, \dots, \|x_{N}\|, \|x_{N-1}\|, \|x_{N}\|+\frac{1}{59} \right\}$ defines a bound on $(x_{n})$. > > > > >- If $\|\cdot\|_{\alpha}$, $\|\cdot\|_{\beta}$ are two [[norm|equivalent norms]] on $X$, then every $\|\cdot\|_{\alpha}$-Cauchy sequence is also a $\|\cdot\|_{\beta}$-Cauchy sequence and conversely. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` # cauchy sequence Suppose $(V, \vert \vert \ \vert \vert)$ is a [[norm]]ed [[vector space]] ([[normed vector space]]) of finite dimension. A [[sequence]] $(v_{n})$ is called a *cauchy sequence* provided that for every $\varepsilon>0$ there exists a [[real numbers]] $N \in \mathbb{R}$ such that whenever $m,\ell \in \mathbb{N_{\geq N}}$ it follows that $\vert \vert v_{\ell} - v_{m} \vert \vert < \varepsilon$. # cauchy criterion Every [[sequence]] in a complete finite dimensional [[inner product space]] [[converge]]s **iff** it is [[Cauchy sequence]] #notFormatted