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Let [itex]S[/itex] be a set with cardinality [itex]|S|=\aleph_0[/itex]. Let [itex]A,B \subseteq S[/itex]. Let [itex]S\backslash A[/itex] and [itex]S\backslash B[/itex] be finite. Then [itex]A \cap B \neq \varnothing[/itex].

How can this be shown? I came across it as an assumption in a proof that a sequence in a metric space can converge to at most one limit.

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# All but finitely many theorem

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