Weierstrass approximation theorem

Weierstrass approximation theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous. For every $\varepsilon>0$ there exists a polynomial $p$ such that

\sup_{x\in[a,b]} |f(x)-p(x)| < \varepsilon.

Equivalently, polynomials are dense in the space of continuous functions on $[a,b]$ with respect to the supremum norm (so $p_n\to f$ uniformly for some polynomial sequence $p_n$ ).

Remarks

A far-reaching generalization is the Stone–Weierstrass theorem, which replaces polynomials by more general subalgebras of continuous functions on compact spaces.