For questions about or using the Weierstrass approximation theorem (or the Stone-Weierstrass theorem). The Weierstrass theorem states that if $f:[a,b]\to\mathbb R$ is continuous and if $\epsilon>0$, then there exists a polynomial $p$ such that $$ |f(x)-p(x)|<\epsilon $$ for all $x$ in $[a,b]$. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.
Convergence of sequences of functions is a major topic in Analysis. The Weierstrass approximation theorem states that the restrictions to $[a,b]$ of the polynomial functions are dense in the space of real-valued continuous functions $C([a,b])$ with respect to the supremum norm. In other words, every $f\in\mathcal{C}\bigl([a,b]\bigr)$ is the uniform limit of a sequence of polynomial functions.
A generalization of this theorem in the context of continuous functions defined on compact topological spaces $X$ is the Stone-Weierstrass theorem. It says that if $A$ is a unital sub-algebra of $C(X,\Bbb R)$ that separates points, then $A$ is dense in $C(X,\Bbb R)$. (Recall that an algebra is a module closed under products, i.e. $f,g\in A,\lambda\in\mathbb R$ implies $\lambda f+g\in A$; a unital sub-algebra of $C(X)$ is an algebra contained in $C(X)$ that contains the constant function $1$.
The theorem is also valid for complex valued functions, if one adds the condition that $A$ is closed under convex conjugation.
