Convergent and Cauchy Sequences

tags:theorem |topology

Convergent and Cauchy Sequences

Statements

Theorem (@Eck23, Theorem 5.1). Any convergent sequence in any metric space is a Cauchy sequence.

Proof

Proof. Let $\{x_n{\}}_{n\in \mathbb{N}}$ be a sequence in the metric space $(X,d)$ and suppose that Let $\{x_n{\}}_{n\in \mathbb{N}}$ converges to $x$. For any $\epsilon >0$, there exists $N>0$ such that, for every $i\ge N$, the inequality $d(x_i,x)<\epsilon /2$ holds. Also, for every $m,n\ge N$, the inequalities $d(x_n,x_m)\le d(x_n,x)+d(x_m,x)<\epsilon /2+\epsilon /2=\epsilon$ hold. Thus, $\{x_n{\}}_{n\in \mathbb{N}}$ is a Cauchy sequence

Remark

An example of a Cauchy sequence that is not convergent. Let $X=\mathbb{Q}$ be the set of rational numbers with the usual metric given by $d(x,y)=|x-y|$. Consider the sequence defined by $x_1=1$, $x_{n+1}=1+1/(1+x_n)$. One can show that this sequence is a Cauchy sequence in $\mathbb{Q}$, but there exists no $x\in \mathbb{Q}$ that is a limit of the sequence.