Does pointwise equicontinuous and uniformly equicontinuous implies compactness?

Question

If every sequence of pointwise equicontinuous functions $M \rightarrow \mathbb{R}$ is uniformly equicontinuous, does this imply that $M$ is compact?

Answer 1

Consider the subspace $M=\mathbb{Z}$ of the reals, an infinite discrete space, certainly not compact. But every sequence of functions is uniformly equicontinuos in that space.