Weierstrass approximation theorem

Let \(a<b\) and let \(f\colon[a,b]\to\CC\) be continuous. For every \(\varepsilon>0\) there exists a polynomial \(p\) such that \[ \sup_{x\in[a,b]}\abs{f(x)-p(x)}<\varepsilon . \] If \(f\) is real-valued, \(p\) may be taken with real coefficients. Equivalently, the polynomials are dense in the Banach space \(C([a,b])\) with the norm \(\norm{g}_\infty\coloneqq\sup_{x\in[a,b]}\abs{g(x)}\).