Weierstrass's Theorem states that any function continuous over a closed interval can be written in terms of a Cauchy sequence of polynomials that converge uniformly to that function. I will not give a proof since the one in front of me is tedious.

Weierstrass's Theorem differs from Taylor's Theorem in a few important ways. While Taylor's Theorem requires analyticity (infinite differentiability) of a function, Weierstrass's Theorem requires only continuity over an interval. However, if a function satisfies the analyticity requirement of Taylor's Theorem, then it can be written as a power series--this means that one gets better and better polynomial approximations by adding higher-order terms, *while leaving the lower-order coefficients fixed*. Weierstrass's Theorem, while more general, does not guarantee a uniformly convergent power series.

Weierstrass's Theorem implies that *any* (e.g. discontinuous) function in Hilbert Space can be written in terms of a sequence of polynomials that converge *in the mean* to that function. Thus Weierstrass's Theorem paves the way for the construction of sets of orthonormal functions that are complete over Hilbert Space (e.g. Legendre Polynomials).