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How can you show that for any infinite set $A$, $|\mathbb{N}|\le |A|$?

thanks

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If you allow the axiom of dependent choice: Just choose distinct elements $a_0,a_1,\dotsc$ in $A$ recursively. This cannot end, since otherwise $A$ would be finite. Thus, $a : \mathbb{N} \to A$ is an injective function.

If you want to stay in ZF, this won't work. There is a difference between infinite sets and Dedekind-infinite sets.

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