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  "body": "\n## Statement\n\nLet $f : \\mathbb{C} \\to \\mathbb{C}$ be an [[entire function|0419794R0T]].\nAssume that $f$ is bounded: there is a real number $M$ with $|f(z)| \\le M$ for each $z \\in \\mathbb{C}$.\nThen $f$ is constant.\n\n## Remarks\n\nProof: let $c \\in \\mathbb{C}$. By [[Cauchy's estimates|0JJRENWMP6]], $|f'(c)| \\le M / r$ for each $r \u003e 0$. So $f'(c) = 0$ for each $c$, and $f$ is constant.\nFor example, $\\sin$ is entire and not constant, so it is not bounded on $\\mathbb{C}$. On $\\mathbb{R}$ it is bounded, which shows that the theorem has no real analogue.\nThe theorem gives a short proof of the [[fundamental theorem of algebra|0JP0W07Y9A]].\n\n## Explanations\n\n### L1 human\n\nA holomorphic function on the whole plane cannot stay inside a bounded region unless it does not move at all.\nEach nonconstant entire function, for example $z$, $e^z$ or $\\cos z$, takes values that are as large as you like.\n",
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