5. Approximation

Approximating continuous functions by polynomials and taming families of functions — the core approximation theorems.

• Arzelà–Ascoli Theorem — When does a family of continuous functions hide a uniformly convergent subsequence? Uniform boundedness plus equicontinuity lets the diagonal argument extract one. Contrast with $x^n$, where it fails.
• Weierstrass Approximation Theorem — Every continuous function on $[0,1]$ is uniformly approximable by polynomials — watch the Bernstein polynomials $B_n(f)$ close in on $f$ as $n$ grows. The fine estimates of the convergence proof are sketched at the idea level.
• Stone–Weierstrass Theorem — A subalgebra that separates points and contains the constants is uniformly dense — trigonometric polynomials on the circle close in on the triangle wave $|x|$, with no algebraic polynomial in sight.
• Müntz–Szász Theorem — Approximate $\sqrt{x}$ by least squares and watch the theorem decide: the dense basis $\{1, x, \ldots, x^N\}$ ($\lambda_n = n$, $\sum \frac{1}{n} = \infty$) shrinks its error $0.1003 \to 0.0708 \to 0.0544$, while the sparse basis $\{1, x, x^4, \ldots, x^{N^2}\}$ ($\sum \frac{1}{n^2} = \frac{\pi^2}{6} < \infty$) stays stuck near $0.16$.