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

Question

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

thanks

See also http://math.stackexchange.com/questions/917593/infinite-set-has-greater-or-equal-cardinality-that-of-n — Gerry Myerson, Sep 03 '14 at 13:21
There are plenty of other duplicates. If someone can be bothered to look for them. — Asaf Karagila, Sep 03 '14 at 13:23

score 3 · Accepted Answer · answered Sep 03 '14 at 13:13

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.

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

1 Answers1

Linked