1.1 Convergent sequences, cauchy sequences, and complete spaces

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Description of convergent sequences and Cauchy sequences in metric spaces. Description of complete spaces.

Convergence

The concept of convergence evaluates whether a sequence of elements is getting “closer” to a given point or not.

Definition 1 Assume a metric space $\left(X,d\right)$ and a countably infinite sequence of elements $\left\{{x}_{n}\right\}:=\left\{{x}_{n},n=1,2,3,...\right\}\subseteq X$ . The sequence $\left\{{x}_{n}\right\}$ is said to converge to $x\in X$ if for any $ϵ>0$ there exists an integer ${n}_{0}\in {\mathbb{Z}}^{+}$ such that $d\left(x,{x}_{n}\right)<ϵ$ for all $n\ge {n}_{0}$ . A convergent sequence can be denoted as ${lim}_{n\to \infty }{x}_{n}=x$ or ${x}_{n}\stackrel{n\to \infty }{\to }x$ .

Note that in the definition ${n}_{0}$ is implicitly dependent on $ϵ$ , and therefore is sometimes written as ${n}_{0}\left(ϵ\right)$ . Note also that the convergence of a sequence depends on both the space $X$ and the metric $d$ : a sequence that is convergent in one space may not be convergent in another, and a sequence that is convergent under some metric may not be convergent under another. Finally, one can abbreviate the notation of convergence to ${x}_{n}\to x$ when the index variable $n$ is obvious.

Example 1 In the metric space $\left(\mathbb{R},{d}_{0}\right)$ where ${d}_{0}\left(x,y\right)=|x-y|$ , the sequence ${x}_{n}=1/n$ gives ${x}_{n}\stackrel{n\to \infty }{\to }0$ : fix $ϵ$ and let ${n}_{0}>⌈1/ϵ⌉$ (i.e., the smallest integer that is larger than $1/ϵ$ ). If $n\ge {n}_{0}$ then

${d}_{0}\left(0,{x}_{n}\right)=|0-{x}_{n}|=|{x}_{n}|={x}_{n}=\frac{1}{n}\le \frac{1}{{n}_{0}}<\frac{1}{⌈1/ϵ⌉}\le \frac{1}{1/ϵ}=ϵ,$

verifying the definition. So by setting ${n}_{0}\left(ϵ\right)>⌈1/ϵ⌉$ , we have shown that $\left\{{x}_{n}\right\}$ is a convergent sequence.

Example 2 Here are some examples of non-convergent sequences in $\left(\mathbb{R},{d}_{0}\right)$ :

• ${x}_{n}={n}^{2}$ diverges as $n\to \infty$ , as it constantly increases.
• ${x}_{n}=1+{\left(-1\right)}^{n}$ (i.e., the sequence $\left\{{x}_{n}\right\}=\left\{0,2,0,2,...\right\}$ ) diverges since for $ϵ<1$ there does not exist an ${n}_{0}$ that holds the definition for any choice of limit $x$ . More explicitly, assume that a limit $x$ exists. If $x\notin \left[0,2\right]$ then for any $ϵ\le 2$ one sees that for either even or odd values of $n$ we have $d\left(x,{x}_{n}\right)>ϵ$ , and so no ${n}_{0}$ holds the definition. If $x\in \left[0,2\right]$ then select $ϵ=\frac{1}{2}min\left(x,2-x\right)$ . We will have that $d\left(x,{x}_{n}\right)\ge ϵ$ for all $n$ , and so no ${n}_{0}$ can hold the definition. Thus, the sequence does not converge.

Theorem 1 If a sequence converges, then its limit is unique.

Proof: Assume for the sake of contradiction that ${x}_{n}\to x$ and ${x}_{n}\to y$ , with $x\ne y$ . Pick an arbitrary $ϵ>0$ , and so for the two limits we must be able to find ${n}_{0}$ and ${n}_{0}^{\text{'}}$ , respectively, such that $d\left(x,{x}_{n}\right)<ϵ/2$ if $n>{n}_{0}$ and $d\left(y,{x}_{n}\right)<ϵ/2$ if $n>{n}_{0}^{\text{'}}$ . Pick ${n}^{*}>max\left({n}_{0},{n}_{0}^{\text{'}}\right)$ ; using the triangle inequality, we get that $d\left(x,y\right)\le d\left(x,{x}_{{n}^{*}}\right)+d\left({x}_{{n}^{*}},y\right)<ϵ/2+ϵ/2=ϵ$ . Since for each $ϵ$ we can find such an ${n}^{*}$ , it follows that $d\left(x,y\right)<ϵ$ for all $ϵ>0$ . Thus, we must have $d\left(x,y\right)=0$ and $x=y$ , and so the two limits are the same and the limit must be unique.

Cauchy sequences

The concept of a Cauchy sequence is more subtle than a convergent sequence: each pair of consecutive elements must have a distance smaller than or equal than that of any previous pair.

Definition 2 A sequence $\left\{{x}_{n}\right\}$ is a Cauchy sequence if for any $ϵ>0$ there exists an ${n}_{0}\in {\mathbb{Z}}^{+}$ such that for all $j,k\ge {n}_{0}$ we have $d\left({x}_{j},{x}_{k}\right)<ϵ$ .

As before, the choice of ${n}_{0}$ depends on $ϵ$ , and whether a sequence is Cauchy depends on the metric space $\left(X,d\right)$ . That being said, there is a connection between Cauchy sequences and convergent sequences.

Theorem 2 Every convergent sequence is a Cauchy sequence.

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