# Usage of Zorn's Lemma to prove that the intersection of all prime ideals contains only nilpotent elements.

I have read a couple proofs that that the intersection of all prime ideals contains only nilpotent elements that use a claim like this:

Suppose that $$a$$ is an element of $$A$$ that is not nilpotent. Let $$S$$ be the set of ideals of $$A$$ that do not contain any element of the form $$a^n$$. Since $$(0) \in S$$, $$S$$ is not empty; then by Zorn's Lemma, $$S$$ has a maximal element $$\mathfrak{m}$$.

However, it's not clear to me that every chain of ideals in $$S$$ has an upper bound, which appears necessary to use Zorn's Lemma. Am I missing something obvious?