I am searching for a metric space in which every Cauchy sequence is stationary.

Question

I am searching for an example of a metric space in which every Cauchy sequence is stationary, means that there exists some $N$ such that for all $n>N$: $a_n = a_N$. Is there a simple example for this?

Answer 1

A necessary and sufficient condition is that the metric space is discrete (i.e. for each $x \in X$ there is $\epsilon > 0$ such that $d(x,y) > \epsilon$ for all $y \ne x$) and complete.