ABSTRACT
This chapter proves a fundamental approximation result for continuous functions called the Weierstrass Approximation Theorem. Fundamentally, it states that a continuous real-valued function defined on a compact set can be uniformly approximated by a smooth function. This is used throughout analysis to prove results about various functions. One can often verify a property of a continuous function by proving an analogous property of a smooth function that is uniformly close to the continuous function. One can write code to implement the Bernstein polynomial approximations on the general interval. The Bernstein polynomials require us to calculate the binomial coefficients. The chapter distinguishes the ideas of a sequence of functions converging at each point in their domain (pointwise convergence) and the idea of uniform convergence. Sequence convergence and integration are described along with limit interchange theorems for integration.
