A metric space $(X, d)$ is said to be totally bounded if for every $\epsilon > 0,$ there is a finite covering of $X$ by $\epsilon$-balls.
Total boundedness implies boundedness.
My question is: does boundedness imply total boundedness?
I'm not able to find a counter example; any hints or a solution would be appreciated.
No. For instance, the space $X=\Bbb N$ endowed with the 0-1 distance is bounded (and complete) but not totally bounded.