Weierstrass theorem

For any continuous real-valued function $f(x)$ on the interval $[a,b]$ there exists a sequence of algebraic polynomials $P_0(x),P_1(x),\dots$ which converges uniformly on $[a,b]$ to the function $f(x)$; established by K. Weierstrass . For generalizations, see Stone–Weierstrass theorem.