I have a question about these metric spaces.
I know that totally bounded -> bounded. And this link gives that totally bounded -> separable. http://www.proofwiki.org/wiki/Totally_Bounded_Metric_Space_is_Separable
So I have that totally bounded -> bounded and separable
But can we go the opposite way, that is can we say for metric spaces that: bounded and separable -> totally bounded?
I know that we can not say separable -> totally bounded, since R is separable, but not totally bounded. And I assume that we can not generally say that bounded -> totally bounded, because then we would have no use for the definition of totally bounded. But if we include both, do we then have a totally bounded metric space?
UPDATE: I got a fast answer that the answer was no, thanks! Then my follow up is this: I really hope you can help me with the follow up aswell.
Would it help if you put in complete aswell? The reason I am asking is because I have read that a subset A for R^n is compact if it is closed and bounded. Genereally a subset A would be compact iff it is closed, totally bounded and complete. So a subset A of R^n that is bounded must then also be totally bounded? What is it with R^n that makes bounded->totally bounded?