Weierstrass Approximation Theorem

Weierstrass Approximation Theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous and let $\varepsilon>0$. Then there exists a polynomial $p$ such that $\sup_{x\in[a,b]} |f(x)-p(x)|<\varepsilon.$

This theorem is foundational in approximation theory: continuous functions can be uniformly approximated by simple algebraic objects (polynomials). It is also a prototype of many “density” results in functional analysis.

Proof sketch: One standard proof uses Bernstein polynomials on $[0,1]$: $B_n(f)(x)=\sum_{k=0}^n f\!\left(\frac{k}{n}\right)\binom{n}{k}x^k(1-x)^{n-k},$ which are polynomials. Uniform continuity of $f$ and concentration properties of the binomial distribution imply $B_n(f)\to f$ uniformly. A linear change of variables transfers the result from $[0,1]$ to $[a,b]$.