# Difference between Zorn's Lemma and the ascending chain condition

Let $$S$$ be a non-empty partially ordered set with respect to a relation $$\leq$$. Then:

Zorn's Lemma: If $$S$$ has the property that any totally ordered subset $$U\subset S$$ has an upper bound, then $$S$$ has a maximal element

Ascending Chain Condition: Every strictly increasing chain eventually stabilises.

The a.c.c implies that every subset of $$S$$ has a maximal element and we can also deduce the same from Zorn's Lemma, so what, if any, is the difference between the two? I feel like they are both saying the same thing

Consider the natural numbers, with an added maximal point which we shall call $\infty$. This partial order satisfies Zorn's lemma, every chain has an upper bound, indeed $\infty$ itself is an upper bound (it's a maximum element!) of any chain.
But the chain $\Bbb N$ is strictly increasing and does not stabilize. So this linear order does not satisfy the ascending chain condition.