TITLE: On the complete metric space
QUESTION [1 upvotes]: 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!

REPLY [3 votes]: 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.
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