Another way to prove completedness in a metric space

Question

Let $X=\mathbb{N}$ and $g(m,n)=0$ if $m=n$, and $g(m,n)=1+\frac{1}{m+n}$ if $m\neq n$. I've proved g is a metric. Is the structure (X,g) complete? sadly I'm not familiar enough with those problems to intuitively deduce it is or not complete. To prove it is complete I'm trying to prove all Cauchy's sequences converge, is this the only way?

Answer 1

It's not hard to show that all Cauchy sequences are eventually constant.

Put differently, points of $X$ are even further apart than with the discrete metric.