Show each infinite set $S \subset \mathbb R$ contains a countably infinite subset

Question

Show each infinite set $S \subset \mathbb R$ contains a countably infinite subset.

I understand that if you remove an object from the set, it will still be infinite, and if we remove another object, it will be countably infinite (I think so at least). I'm very lost on this kind of proof. However if someone can help that would be appreciated.

Each incite set $S$ contains a countable subset. Should $S$ be a subset of $\mathbb R$ or not. — mathcounterexamples.net, Sep 08 '20 at 17:53
@mathcounterexamples.net: I suppose you meant Each infinite set — J. W. Tanner, Sep 08 '20 at 18:00
Since all finite sets are countable, this is trivial as stated. I expect that what you really want to prove is that every infinite subset of $\Bbb R$ contains a countably infinite subset. — Brian M. Scott, Sep 08 '20 at 18:00
This is a restatement of the countable axiom of choice (aka $\omega$-AC). This equivalent to saying: $S$ has properly subset with the same cardinal as $S$. — miraunpajaro, Sep 08 '20 at 18:00
Some authors use countable set to mean countably infinite set — J. W. Tanner, Sep 08 '20 at 18:04
@BrianM.Scott I am going to go ahead and make that revision, and I’ll also include [tag:induction] tag based on Aman’s answer — gen-ℤ ready to perish, Sep 08 '20 at 18:15
@gen-ℤreadytoperish: That isn’t actually an instance of induction as a proof technique: it’s a recursive construction. — Brian M. Scott, Sep 08 '20 at 18:25
@J.W.Tanner: I know. I also consider it unfortunate; at the very least people should be made aware of the more common usage. — Brian M. Scott, Sep 08 '20 at 18:28
Does this answer your question? Prove that every infinite set has a countable subset. — Physical Mathematics, Sep 09 '20 at 16:23

score 3 · Answer 1 · answered Sep 08 '20 at 17:56

Clearly $S$ is nonempty so there is an element $a_1 \in S$. Then $S -\{a_1\}$ can’t be empty (as $S\neq \{a_1\} $since $S$ is not finite) and so we can pick $a_2 \in S -\{a_1\}$. Next pick $a_3 \in S - \{a_1, a_2\}$ inductively you will get a sequence and you are done!

Show each infinite set $S \subset \mathbb R$ contains a countably infinite subset

1 Answers1