Discrete metric spaces are complete

Result: Discrete metric spaces are complete.
Proof:
          Let $\{x_n\}$ be a Cauchy sequence in a discrete metric space $(X,d)$, then given $\epsilon > 0$ there exists a positive integer $p \in \mathbb{N}$ such that $d(x_n , x_m) < \epsilon$ for all $m,n \ge p$. Let $\epsilon = 1/2$ then there exists a positive integer $p \in \mathbb{N}$ such that $d(x_n , x_m) < 1/2$ for all $m,n \ge p$. But $d(x_n , x_m) < 1/2 \to d(x_n , x_m) = 0$ for all $m,n \ge p$ in discrete metric. Hence $x_n = x_m = a (say)$ for all $m,n \ge P$ and hence this cauchy sequence $\{x_n\}$ is an eventually constant sequence $a$ and converge to $a$. This completes the proof.

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