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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be continuous.\nThen for every $\\varepsilon \u003e 0$ there is a polynomial $p$ with $|f(x) - p(x)| \u003c \\varepsilon$ for all $x \\in [a, b]$.\n\n## Remarks\n\nEquivalently, there is a sequence of polynomials that converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.\nOne proof uses the [[Bernstein polynomials|0C4VWBFY73]] on $[0, 1]$, together with [[uniform continuity|0W0TZDRF9C]] of $f$.\nThe Stone–Weierstrass theorem generalizes the result to compact spaces. Mathlib states the theorem as `polynomialFunctions_closure_eq_top`.\n\n## History\n\nKarl Weierstrass proved the theorem in 1885.\n\n## Explanations\n\n### L1 human\n\nEvery continuous function on a closed bounded interval is uniformly close to a polynomial.\nSo polynomials can imitate any continuous graph as closely as we like.\n",
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