Weierstraß Approximation Theorem

If $f$ is a continuous real-valued function on $[a,b]$ and if any $\epsilon > 0$ is given, then there exists a Polynomial $p$ on $[a,b]$ such that

$$\vert f(x)-P(x)\vert<\epsilon$$

for all $x\in[a,b]$. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by Polynomials to any degree of accuracy.

See also Müntz's Theorem