Incomplete metric on $\mathbb{N}$, set of Natural Number

Question

$\mathbb{N}$ is complete with respect to usual metric.

But if I define $d(x, y) =|\frac{1}{x} - \frac{1}{y}|$ then $\mathbb{N}$ is incomplete. How to show this? It is quite interesting.

I thought about a cauchy sequence which is not convergent but unable to do that.

Answer 1

Hint: Every strictly increasing sequence is Cauchy with this metric.

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[Additional Comment: In fact, a sequence $(a_n)$ is Cauchy in this space if and only if $a_n\to\infty$.]

[Additional Additional Comment: My previous comment was wrong (thanks, @Mars Plastic). Striking it out. ]