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.

Consider the sequence $1,2,3,\ldots$. – David Mitra Aug 01 '19 at 13:40 — David Mitra, Aug 01 '19 at 13:40

MPW · Accepted Answer · 2019-08-01T14:09:54.537

3

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

~~[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. ]

edited Aug 01 '19 at 14:09

answered Aug 01 '19 at 13:45

MPW

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Your additional comment isn't quite true, as the constant sequences obviously converge. – Mars Plastic Aug 01 '19 at 14:06
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@MarsPlastic : You are correct, of course. Editing. +1 for you. – MPW Aug 01 '19 at 14:08
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On the other hand $(a_n)$ is Cauchy and divergent if and only if $a_n \to \infty$. – Theo Bendit Aug 01 '19 at 14:16

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