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!