6. Approximation

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

• Arzelà–Ascoli Theorem — When can we pull a uniformly convergent subsequence out of a family of continuous functions? Two conditions do the job. First, every function stays inside one 📦 box. Second, all of them wiggle with similar gentleness. A diagonal argument over the rationals then builds a subsequence. It closes in everywhere at once. The power functions fail the second condition: they turn steep suddenly near 1. For example, picture 30 🧑 friends walking across a playground. Everyone must stay inside the fence. Nobody may suddenly sprint. Then you can pick a group whose walks grow alike. One friend sprinting at the end ruins the group.
• Weierstrass Approximation Theorem — Can polynomials alone imitate every continuous function on an interval? A Bernstein polynomial averages the function's values near each point. As the degree grows, the average concentrates near that point. So the polynomial follows even the bumpy corners. The proof's fine estimates are only sketched. For example, think of replacing a slope with a 🪜 staircase. The finer the steps, the smaller the gap anywhere you measure. Raising the degree presses the polynomial onto the whole function at once.
• Stone–Weierstrass Theorem — Can materials other than polynomials approximate every continuous function? The ingredient set needs three things. Closure under addition and multiplication. The power to tell two different points apart. And the constant functions. With only these, it tracks any continuous function as closely as you like. For example, layering round waves can build even a pointy 🏠 roof shape. In this scene, waves of sines and cosines close in on the triangle wave on a circle.
• Müntz–Szász Theorem — Can we skip some powers and still approximate everything? The test is the sum of the reciprocals of the exponents. If that sum grows forever, the ingredients are dense enough. If it stays finite, some function remains forever out of reach. The scene fits the square-root function by least squares and compares a dense set with a sparse one. For example, think of measuring with a 📏 ruler. Marks packed endlessly close let you measure any length well. Marks that thin out like 1, 4, 9 leave some lengths unmeasurable forever.