Here's another one I like. Technically it's a calculus question, but you can make it Analysis by justifying each of the steps involved. You'll need at least one major theorem.
Suppose f:[a,b]->R is bounded and continuous except a finite number of points z_1, ..., z_k. Prove f:[a,b]->R is riemann integrable. (I believe this is also true if f is continuous except at a countable number of points)
I'm not sure my proof is correct but I'd love to see yours if you come up with one so we can compare.