Convolutions and Polynomial Approximation

In this video, I intuitively explain and apply some deeper mathematical tools - namely convolutions and approximate identities - to prove the Weierstrass approximation theorem, which roughly states that any continuous function can be approximated by polynomials. I also make connections to other examples where approximate identities show up, like the Gaussian blur and moving averages. Some familiarity with calculus is recommended, but not required. Submitted for the 3Blue1Brown Summer of Math Exposition 1 contest.