On the complete metric space

Question

Let $\mathbb N$ be the set of all natural numbers. Set $$d(m, n) = \left\{\begin{array}{ll} 0, &\text{if } m = n, \\ 1+ \dfrac{1}{m+n}, &\text{if }m\neq n. \end{array}\right.$$

Prove that $(\mathbb N, d)$ is the complete metric space.

Thanks all for help!

Answer 1

Since $d(m,n)\gt1$ whenever $m\neq n$, the only Cauchy sequences are those that are eventually constant. But a sequence that is eventually constant converges. Thus $(\Bbb N,d)$ is complete.