
Counterexample
I had an analysis exam today, and the last question was: Give an example of a set $\displaystyle F\subset\mathcal{C}_b(\mathbb{R},\mathbb{R})$ that is pointwise compact, (uniformly) equicontinuous, and NOT totally bounded.
I know $\displaystyle F$ has to be open, otherwise by the ArzelaAscoli 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 $\displaystyle F=\{n\sin(x/n)~~n\in\mathbb{N}\}$. I am pretty sure this doesn't work, so what would an example of a function with the above properties be?
Note that $\displaystyle \mathcal{C}_b(\mathbb{R},\mathbb{R})$ is the space of all bounded functions from the reals to the reals.