Let $X$ be a set, and let $d:X\times X\to\mathbb{R}$ be the [discrete metric](/page/Discrete%20Metric), meaning $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\ne y$. A sequence $x:\mathbb{N}\to X$ is Cauchy in $(X,d)$ if and only if there exist $N\in\mathbb{N}$ and $a\in X$ such that $x_n=a$ for every $n\ge N$.