# Prove that for any infinite set $A$, $|\mathbb{N}|\le |A|$ [duplicate]

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

thanks

• – Gerry Myerson Sep 3 '14 at 13:21
• There are plenty of other duplicates. If someone can be bothered to look for them. – Asaf Karagila Sep 3 '14 at 13:23
• Sorry, my bad. :( – maxuel Sep 3 '14 at 13:47

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.