Proof of Zorn's Lemma using the Axiom of Choice. Why is $\mathscr U$ a tower?

Solution 1:

Either $C$ contains the union, or it does not. If it doesn't, then at least some element of the union lies outside $C$. Such an element must be contained in at least one of the sets in the chain. Let $B$ be such a set in the chain. Then $B$ does not lie in $C$, so $g(C)$ lies in $B$, so $g(C)$ lies in the union.

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