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.


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