# Approximation theorems and sketches of their proofs

I would like to collect (with support of users here) approximation theorems and sketches of their proofs.

Each answer would give one approximation theorem and sketch of a proof. Any other proof of the same theorem by other users is expected to be added to the one proof that is already there.

Some examples of approximation theorems:

1. Any continuous function can be approximated by polynomial functions. More precisely, suppose  $$f$$  is a continuous real-valued function defined on the real interval $$[a, b]$$. For every $$\epsilon> 0$$, there exists a polynomial $$p$$ such that for all $$x\in [a, b]$$, we have $$| f (x) − p(x)| < \epsilon$$, or equivalently, the supremum norm $$|| f − p||_{\infty} < \epsilon$$. This is called Weierstrass Approximation Theorem.
2. Any smooth function can be approximated by polynomials in the sense of Taylor series expansion. I do not know if there is any name for this.
3. Let $$M$$ be a real analytic manifold and $$U$$ an open set of $$M$$. Let $$f : U \to \mathbb{R}$$ a continuous function. Then, there is a smooth function $$g : U \to \mathbb{R}$$ and a constant $$C>0$$ such that $$|f(x)-g(x)| \leq C, \quad \forall x \in U.$$

Please share at least one theorem and sketch of a proof. There is no restriction on the level of difficulty of the theorem. Any level is welcome.