Are some discrete (and all finite) metric spaces complete?

For example, it seems to me from the definition of complete that $\mathbb{N}$ with (say) the Euclidean metric would be complete, since any Cauchy sequence on $\mathbb{N}$ must converge to an integer. (That is, it would look like 5,1,4,2,3,3,3,3,3,3....) So is it actually complete?

Similarly, it seems that any metric space on a finite set would be complete as well. Is this correct, and if not, why not?

If you have suggestions for texts or online resources that would help me with these types of definitions, that would also be awesome. Thanks very much!

Solution 1:

You are correct in your reasoning. More generally, if $(M, d)$ is a metric space such that there exists $\epsilon > 0$ such that for any $x, y \in M$, we have $d(x,y) > \epsilon$ whenever $x\neq y$, then $M$ is complete.

To see this, say $\{x_n\}$ is a Cauchy sequence. Then there exists $N\in\mathbb{N}$ such that for all $n, m \geq N$, $d(x_n, x_m) < \epsilon$ (same $\epsilon$ as above). But if $x_n\neq x_m$, then $d(x_n, x_m) > \epsilon$. In other words, for all $n, m \geq N$, $x_n = x_m$ - that is, the "tail" of the sequence is constant. Hence the sequence obviously converges.