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  "body": "\n## Statement\n\nLet $A$ be an infinite set, let $B$ be a finite set, and let $f : A \\to B$ be a function.\nThen there is an element $b \\in B$ such that the fibre $f^{-1}(b)$ is infinite.\n\n## Remarks\n\nThe proof is short. If each of the finitely many fibres were finite, then $A$, their union, would be finite.\nExample: in any infinite sequence of integers, infinitely many terms have the same remainder modulo $10$.\nThe statement is the starting point of the infinite form of [[Ramsey's theorem|00TQV3F5S4]].\n\n## Explanations\n\n### L1 human\n\nIf you put infinitely many objects into finitely many boxes, at least one box gets infinitely many of them.\nFor example, colour each natural number red or blue. Then one of the two colours is used infinitely often.\n",
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