Lecture 21: Integrals and Derivatives under Uniform Convergence

We use Weierstrass M-test to prove continuity of power series inside the radius of convergence. We also show that C([a; b]) with the supremum norm is Cauchy complete. This will later play an important role when we establish existence and uniqueness of solutions to ordinary differential equations. Toward the end of the lecture we show that integrals and differentiation behave well under uniform convergence.