Problem Statement: Let be uniformly continuous and let for every . Show converges uniformly.
Proof: First I claim that converges pointwise to .
Let . Then . Since is continuous and converges to it follows that . This holds for all and so converges to pointwise.
Now to show that this convergence is uniform. Let . Since is uniformly continuous there exists a such that for every it follows that .
Let , then there exists a such that for every it follows that and so .
For all and for every it follows that . Uniform continuity implies that for all and for all . But , thus, for every and for all . Therefore, converges to uniformly on .
Reflection: The main thing in this proof is that we know is uniformly continuous. It makes sense that the convergence has to be uniform. It convergence was only pointwise then your function could be doing crazy things and so wouldn’t be uniformly continuous.