Let $C$ be the space of continuous functions $K \to M$, where $K$ is a compact space, $M$ is a compact metric space. Equip $C$ with the obvious metric. The Arzelà-Ascoli theorem says that any equicontinuous family $F \subset C$ is relatively compact. One can do so by appealing to Lebesgue's number lemma to show that its closure is totally bounded.
I want to study extent to which the Arzelà-Ascoli fails if we drop some of its hypotheses. For instance, is there an example of a sequence in $C$ that is (a) Cauchy, (b) pointwise convergent, but (c) not convergent to a point in C? (Obviously, the points of any such sequence are not contained in any equicontinuous family.)