First, replacing by , we may assume that , so that for all in .

Next, it follows from the uniform boundedness principle that is bounded, say for all .

Now let . For each in , there exists such that whenever . Let denote the open set . Then for all in , whenever . Such sets form an open cover of a given compact subset of , and by looking at a finite subcover you should be able to see that the pointwise convergence is uniform on that set.