is a space with discrete metric space complet or not?

Question

I have a question about discrete metric spaces:

prove that Every discrete metric space $X$ with discrete metric, ($X$,$d_0$) is complete?

Answer 1

If $(X, d_0)$ is a metric space such that there exists $\epsilon > 0$ such that for any $x, y \in X$, we have $d_0(x,y) > \epsilon$ whenever $x\neq y$, then $E$ is complete.

Proof: Let $\{x_n\}$ be a Cauchy sequence. $\exists N\in\mathbb{N}\text{ such that }\forall n, m \geq N, d_0(x_n, x_m) < \epsilon$. If $x_n\neq x_m$, then $d_0(x_n, x_m) > \epsilon$. $\forall n, m \geq N, x_n = x_m$ - the sequence thus obviously converges. Q.E.D.