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I am using lee larson's lecture note on introductory real analysis for self learning. And I am stucking in one problem which is like this:

If $S$ is an infinite set, then there is a countably infinite collection of nonempty pairwise disjoint infinite sets $T_{n}$$(n\in \mathbb{N})$ such that $S=\cup_{n\in \mathbb{N}}T_{n}$

The definition of infinite set in the context is just "not finite". And the only tool he gives before is Cantor-Berstein theorem. And he even didin't prove that every infinite set has a countable infinite subsets. The exercise just before this is tring to prove that infinite is equivalent with it has a bijection to its proper subset.

Maybe the auther gives not enough tools for this question, I have no idea about this problem. I try to find a constructive proof, but I failed, I can't prove that if I operate the difference(i.e. $S-T$, $T$ is a infinite proper subset of $S$) then the result is also infinite. Hope that anyone can help me with this.

UESTCfresh
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  • https://math.stackexchange.com/questions/12629/partitioning-an-infinite-set – Eric Towers Jan 22 '22 at 03:49
  • Proving that an infinite set has a countably infinite subset is easy: "pick an element, then pick another element etc" , this process can't stop as the set would then be finite. Now , say $A_n$ be the singleton consisting of the $n$th element of countably infinite set. Then group the elements that remain into another set (say $B$ ) and rename everything to get the $T_n$s . Say $T_1=B$ and $T_{n+1}=A_n$ otherwise. – aritracb Jan 22 '22 at 03:54
  • @EricTowers The link you give can not make me clearer. The methods used there are out of my range of knowledge. – UESTCfresh Jan 23 '22 at 03:50

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