So, as the title says I'd like to give a proof of the fact that if $C$ is an infinite set then it is equipotent or equivalent to its cartesian product $C\times C$ using Zorn's Lemma (and of course some of its implications like the fact that $C$ has an infinite countable subset which I think might be very useful).
The main problem that I have is that I'm not supposed to use any theorem or result involving cardinal numbers since I'm still taking an elementary set theory course which hasn't covered that topic yet and all the proofs that I've read so far use cardinals arithmetic at some point.
Another thing that I think might be useful is a lemma which was proved via Zorn's Lemma in an answer to the question Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$ which states that given the infinite set $C$ there exists a non-empty set $B$ such that $B\times \mathbb{N}$ is equipotent to $C$. Then it suffices to give a bijection from $(B\times \mathbb{N})$ $\times$ $(B\times \mathbb{N})$ to $B\times \mathbb{N}$
So, any suggestion on the direct proof (without cardinals, unfortuntely) via Zorn's Lemma or an actual bijection from $(B\times \mathbb{N})$ $\times$ $(B\times \mathbb{N})$ to $B\times \mathbb{N}$ would be highly appreciated. Thanks in advance.
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