There is a famous proof of the existence of a transcendental number by comparison of cardinalities.
I would like to know more examples of existence proof by comparison of cardinalities.
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See my answers to Does the set of entire functions have the same cardinality as the reals? AND Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold? There are probably a lot more examples in MSE by others. – Dave L. Renfro May 17 '23 at 08:42
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See also: Looking for a problem where one could use a cardinality argument to find a solution. and What are interesting examples of existential proofs based on cardinality arguments? – Martin Sleziak May 18 '23 at 07:25