Problem Statement: Let . Prove is Riemann Integrable on and .
Proof: First note that each is continuous on and since is a sum of continuous functions it follows that is continuous on . Furthermore, since is a compact set it follows that this convergence is uniform. Since we have uniform convergence we get the following:
Reflection: At first glance this problem scared me. But, after being reminded by Suz that convergence of a sum to a function on a compact set gives us uniform convergence my life was made easier. This is another one of those handy facts to remember.