Set that is bounded but not totally bounded: Reading textbook

I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally bounded. This makes sense, however I am having trouble thinking of an example of a set that is bounded but not totally bounded. Could anyone shed some light on this? Thanks!

I know this is an old post, but any infinite set M with a discrete metric is bounded by any N>1 but it is not totally bounded for open balls with $\epsilon\leq 1$ because for it to be totally bounded it must have a finite number of points. But such a ball is only a singleton, and you would require infinitely many to cover M.

Take $U=\{e_n|\:n\in\mathbb N\}\subset \ell^\infty (\mathbb R)$.

It is obviously bounded since $\forall x\in U \:\|x\|=1$, but $\forall x,y\in \ell^\infty$ we have $d(x,y)=1$, so obviously for $\epsilon=1$ there is no finite number of open balls with radius $\epsilon$ that cover $U$ - cause each ball would contain at most one member of $U$.