## Counterexample

I had an analysis exam today, and the last question was: Give an example of a set $F\subset\mathcal{C}_b(\mathbb{R},\mathbb{R})$ that is pointwise compact, (uniformly) equicontinuous, and NOT totally bounded.

I know $F$ has to be open, otherwise by the Arzela-Ascoli theorem, it would be compact and therefore totally bounded, but unfortunately, I got to this question with ~3 minutes left, and was only able to come up with $F=\{n\sin(x/n)~|~n\in\mathbb{N}\}$. I am pretty sure this doesn't work, so what would an example of a function with the above properties be?

Note that $\mathcal{C}_b(\mathbb{R},\mathbb{R})$ is the space of all bounded functions from the reals to the reals.