In a metric space are the terms $bounded$ and $totally$ $bounded$ interchangeable?

Question

The question is: Is there a theorem that says in a metric space a set is bounded if and only if it is totally bounded?

Answer 1

No. Consider $\mathbb N = \{1,2,3,\cdots\}$ with the discrete metric $d(x,y)=0,1$ according to whether or not $x=y$.

This cannot be covered with a finite number of $\epsilon=\frac 1 2$ balls, but no two points are more than $1$ from each other.

Answer 2

No. On any infinite set, let the distance between two distinct members be $1$. That's a metric space that is bounded but not totally bounded.