This is a question I was asked on my oral exam for set-theory. I found it a particular interesting one:

Show that:

For any infinite set $\displaystyle X$, there exists a bijection $\displaystyle f:X\times X \to X \Leftrightarrow$ well-orderings-theorem of Zermelo.

It's in particular the implication $\displaystyle \Longrightarrow $ that's interesting.

Hint 1.

Spoiler:

Addition hint 1.

Spoiler: