1.1 Convergent sequences, cauchy sequences, and 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 $(X,d)$ 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 $ϵ>0$ there exists an integer $n_0\in {\mathbb{Z}}^{+}$ such that $d(x,x_n)<ϵ$ 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 $(\mathbb{R},d_0)$ where $d_0(x,y)=|x-y|$ , the sequence $x_n=1/n$ gives $x_n\stackrel{n\to \infty }{\to }0$ : fix $ϵ$ and let $n_0>\lceil 1/ϵ\rceil$ (i.e., the smallest integer that is larger than $1/ϵ$ ). If $n\ge n_0$ then

verifying the definition. So by setting $n_0\left(ϵ\right)>\lceil 1/ϵ\rceil$ , we have shown that $\left\{x_n\right\}$ is a convergent sequence.

Example 2 Here are some examples of non-convergent sequences in $(\mathbb{R},d_0)$ :

• $x_n=n^2$ diverges as $n\to \infty$ , as it constantly increases.
• $x_n=1+{(-1)}^n$ (i.e., the sequence $\left\{x_n\right\}=\{0,2,0,2,...\}$ ) 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 $x\notin [0,2]$ then for any $ϵ\le 2$ one sees that for either even or odd values of $n$ we have $d(x,x_n)>ϵ$ , and so no $n_0$ holds the definition. If $x\in [0,2]$ then select $ϵ=\frac{1}{2}min(x,2-x)$ . We will have that $d(x,x_n)\ge ϵ$ 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 $d(x,x_n)<ϵ/2$ if $n>n_0$ and $d(y,x_n)<ϵ/2$ if $n>n_0^{\text{'}}$ . Pick $n^{*}>max(n_0,n_0^{\text{'}})$ ; using the triangle inequality, we get that $d(x,y)\le d(x,x_{n^{*}})+d(x_{n^{*}},y)<ϵ/2+ϵ/2=ϵ$ . Since for each $ϵ$ we can find such an $n^{*}$ , it follows that $d(x,y)<ϵ$ for all $ϵ>0$ . 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.

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 $d(x_j,x_k)<ϵ$ .

As before, the choice of $n_0$ depends on $ϵ$ , 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.