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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 $(X,d)$ and a countably infinite sequence of elements $\left\{{x}_{n}\right\}:=\{{x}_{n},n=1,2,3,...\}\subseteq X$ . The sequence $\left\{{x}_{n}\right\}$ is said to converge to $x\in X$ if for any $\u03f5>0$ there exists an integer ${n}_{0}\in {\mathbb{Z}}^{+}$ such that $d(x,{x}_{n})<\u03f5$ 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 $\u03f5$ , and therefore is sometimes written as ${n}_{0}\left(\u03f5\right)$ . 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 $(\mathbb{R},{d}_{0})$ where ${d}_{0}(x,y)=|x-y|$ , the sequence ${x}_{n}=1/n$ gives ${x}_{n}\stackrel{n\to \infty}{\to}0$ : fix $\u03f5$ and let ${n}_{0}>\lceil 1/\u03f5\rceil $ (i.e., the smallest integer that is larger than $1/\u03f5$ ). If $n\ge {n}_{0}$ then
verifying the definition. So by setting ${n}_{0}\left(\u03f5\right)>\lceil 1/\u03f5\rceil $ , we have shown that $\left\{{x}_{n}\right\}$ is a convergent sequence.
Example 2 Here are some examples of non-convergent sequences in $(\mathbb{R},{d}_{0})$ :
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 $\u03f5>0$ , and so for the two limits we must be able to find ${n}_{0}$ and ${n}_{0}^{\text{'}}$ , respectively, such that $d(x,{x}_{n})<\u03f5/2$ if $n>{n}_{0}$ and $d(y,{x}_{n})<\u03f5/2$ if $n>{n}_{0}^{\text{'}}$ . Pick ${n}^{*}>max({n}_{0},{n}_{0}^{\text{'}})$ ; using the triangle inequality, we get that $d(x,y)\le d(x,{x}_{{n}^{*}})+d({x}_{{n}^{*}},y)<\u03f5/2+\u03f5/2=\u03f5$ . Since for each $\u03f5$ we can find such an ${n}^{*}$ , it follows that $d(x,y)<\u03f5$ for all $\u03f5>0$ . Thus, we must have $d(x,y)=0$ and $x=y$ , and so the two limits are the same and the limit must be unique.
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 $\u03f5>0$ there exists an ${n}_{0}\in {\mathbb{Z}}^{+}$ such that for all $j,k\ge {n}_{0}$ we have $d({x}_{j},{x}_{k})<\u03f5$ .
As before, the choice of ${n}_{0}$ depends on $\u03f5$ , and whether a sequence is Cauchy depends on the metric space $(X,d)$ . 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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