To check Completeness and totally boundedness of a metric space.

Question

Let$X=N$ be set of positive integers.Consider the metric d such that. $d(m,n)=\frac{1}{m}-\frac{1}{n}$.Then how to verify whether $(X,d)$ is complete and totally bounded?
We know that only Cauchy sequence in N is constant.I don't understand how to think further.Thank you.

Answer 1

It is not true that Cauchy sequences are constants. In fact $\{1,2,\cdots\}$ is itself a a Cauchy sequence in this space. Also this sequence does not converge, so the space is not complete. The space is totally bounded. Let $\epsilon >0$ and choose $N$ such that $\frac 2 N <\epsilon$. Let us verify that $\{1,2,\cdots,N\}$ is an $\epsilon-$ net. Clearly any integer $\leq N$ is at distance at most $\epsilon$ from some member of $\{1,2,\cdots,N\}$. Consider an integer $n >N$. Then $d(n,N) =|\frac1 n -\frac 1 N| \leq \frac 1 n+\frac1 N \leq 2\frac 1 N <\epsilon$.