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.