I understand that the crux of the proof is that uniform continuous function on compact set is uniform convergence. However, I do not understand why we need a middle space $W$, isn't that working with $V$ enough? What would happen if do not add such a $W$ in between $V$ and $U$? I was told that, the idea of having $W$ is that, if we can show convergence in $W$, then automatically we can show convergence in a smaller space.
This is intuitive. But I am not convinced mathematically and I wish to see what would happen without $W$ in the proof.