# On chain conditions and Zorn's Lemma, again

I'm doing a introductory course on commutative algebra, and have just been introduced to the chain conditions. I know there's a lot of questions probably similar, but I haven't really understood this. In an online lecture on chain conditions, the following Proposition was proved:

Prop 1: TFAE

1. Every chain $$a_1\leq a_2\leq a_3\dots$$ in a poset $$\Sigma$$ terminates, i.e. $$\exists n_0$$ s.t. $$a_n=a_{n+1}, \quad \forall n\geq n_0$$.
2. Every non-empty subset of $$\Sigma$$ has a maximal element.

Then we have Zorn's:

Zorn's Lemma: Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.

Question: To me it then seems as if Prop 1 $$\Rightarrow$$ Zorn's Lemma, since the first part of Zorn's seems imply 1 of Prop 1. And the whole of $$P$$ is certainly a non-empty subset of $$P$$, so it has a maximal element by Prop 1. What have I misunderstood?

I did read this thread, but I didn't get any wiser. I think my problem understanding this is even more fundamental/trivial; But perhaps I missed something.

EDIT: Or is it just that perhaps it isn't true that part 1 of zorn's imply 1 of Prop 1? Or something else/more?

For instance, consider the ordinal $$\omega+1$$ as an ordered set. It has a greater element, $$\omega$$. However, it has an infinite increasig sequence given by $$a_n=n$$.