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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 1}$ and $(b_n)_{n \\ge 1}$ be real numbers.\nLet $E \\subset \\R$ be a measurable set of positive [[Lebesgue measure|2Z2QVWZ262]].\nAssume that $a_n \\cos nx + b_n \\sin nx \\to 0$ as $n \\to \\infty$ for every $x \\in E$.\nThen $a_n \\to 0$ and $b_n \\to 0$.\n\n## Remarks\n\nProof idea: write $a_n \\cos nx + b_n \\sin nx = \\rho_n \\cos(nx + \\varphi_n)$ with $\\rho_n = \\sqrt{a_n^2 + b_n^2}$.\nIf $\\rho_n \\not\\to 0$, then $\\rho_{n_j} \\ge \\varepsilon \u003e 0$ along a subsequence, so $\\cos^2(n_j x + \\varphi_{n_j}) \\to 0$ on $E$.\nReplace $E$ by a bounded subset of positive measure.\nBy [[dominated convergence|BYNV84PF86]], $\\int_E \\cos^2(n_j x + \\varphi_{n_j}) \\, dx \\to 0$.\nBut this integral equals $\\frac{1}{2} \\abs{E} + \\frac{1}{2} \\int_E \\cos(2 n_j x + 2 \\varphi_{n_j}) \\, dx$, and the last integral tends to $0$ by the [[Riemann–Lebesgue lemma|WVKDC011KP]], a contradiction.\n\nIn complex form: if $c_n e^{inx} + c_{-n} e^{-inx} \\to 0$ for every $x$ in a set of positive measure, then $c_n \\to 0$ as $\\abs{n} \\to \\infty$.\nSo a [[trigonometric series|FZK2M8KGXQ]] that converges on a set of positive measure has coefficients tending to $0$.\n\nExample: if $\\sum_n (a_n \\cos nx + b_n \\sin nx)$ converges for every $x$ in $[0, 1]$, its terms tend to $0$ there, so $a_n, b_n \\to 0$.\n\nPositive measure is needed: at the single point $x = 0$ the terms $n \\sin nx$ are all $0$, but $b_n = n \\not\\to 0$.\n\n## Explanations\n\n### L1 human\n\nIf a sequence of waves with amplitudes $\\rho_n$ dies out at every point of a set of positive length, then the amplitudes themselves must go to $0$.\nA wave of large amplitude cannot be small on a large set, because it oscillates and spends a fixed fraction of the time far from $0$.\n\n### L2 human\n\nThe theorem is the first step in the proof of [[Cantor's uniqueness theorem|NSS3XW4BYZ]].\nIt reduces questions about convergent trigonometric series to series whose coefficients tend to $0$, where Riemann's methods apply.\nIts proof is a typical use of the Riemann–Lebesgue lemma together with a convergence theorem for integrals.\n",
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