Is it a Cauchy sequence?

Question

Let $(X,d)$ be a metric space and $(x_n)$ is a sequence in $X$. Then
$\sup_{p\ge 1} d(x_n,x_{n+p}) \rightarrow 0$ as $n\rightarrow \infty$ implies $(x_n)$ is a Cauchy sequence.
Any hints or counterexample to prove or disprove this implication? Thanks in advance.

Answer 1

Given $\epsilon>0$ there is $N$ such that $\sup_{p\ge 1}d(x_nx_{n+p})<\epsilon$ for all $n>N$. Then for all $m>n>N$, we have $d(x_n,x_m)\le \sup_{p\ge 1}d(x_n,x_{n+p})<\epsilon$.