Lemma 2

If T is a locally compact space with a countable base, then the Banach space $\mathscr{K}(Y)$ of continuous numerical functions tending to 0 at the point at infinity is separable.

This lemma is none other than the Cor. to Th. 1 of GT, X, §3, No. 3. One may observe that it also follows from Lemma 1 and the fact that the topology of uniform convergence on $\mathscr{K}(Y)$ is coarser than the direct limit topology of the topologies of the subspaces $\mathscr{K}(Y, \overline{U}_n)$.