Weierstrass Approximation Theorem

The Weierstrass Approximation Theorem states that any continuous function defined on a closed interval can be closely approximated by a sequence of polynomial functions. In simple terms, it means that even complex, smooth curves can be approached arbitrarily well using just sums of powers like quadratics, cubics, etc. This is important because polynomials are well-understood and easy to work with, so it's powerful to know they can approximate a wide variety of functions as closely as needed within a specific range.