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}^{b}f(x)x^{n}dx=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 f^{2}. Hence:

∫_{a}^{b}(f(x))^{2}dx=lim∫_{a}^{b}f(x)p_{n}dx=0, and thus f=0 as required. This is called the moments problem.