Weierstrass approximation theorem

The Weierstrass approximation theorem states that for any continuous function f on a closed bounded interval, there exists a sequence of polynomials on the same interval converging uniformly to f. I'll node the proof here. This is termed a "uniform approximation".

The theorem is incredibly useful for dealing with limits concerning continuous functions - as the approximation is uniform, it gives us much more information about limiting processes. Here's a simple application. Suppose that, for a given closed bounded interval from a to b and a given continuous function f from the interval to ℝ, we have that for every n:

∫_a^bf(x)xⁿdx=0.

We want to show that f=0. Now we can construct a sequence p_n of polynomials converging uniformly to f. Since f is bounded, fp_n converges uniformly to f². Hence:

∫_a^b(f(x))²dx=lim∫_a^bf(x)p_ndx=0, and thus f=0 as required. This is called the moments problem.