How do you prove that a set doesn't contain maximal elements? I know that the Zorn's lemma (If every totally ordered subset of a partially ordered set S has an upper bound, then S contains a maximal element) can be used to prove the existence of a maximal element.
However, I need to prove that a certain partially ordered set S, which contains infinite (countable) elements, has no maximal element.
Now I suppose that the reverse of Zorn's lemma is not necessarily true (that is: If a partially ordered set S contains a maximal element, then every totally ordered subset of S has an upper bound). Mine is just a supposition because I cannot figure out a counterexample, but my gut instinct - and the fact that I always found Zorn's lemma statement written as an "if - then" and never as an "if and only if" - is telling me that the reverse is not true.
Now I now that the Zorn's lemma is equivalent to: "If a partially ordered set S doesn't contain a maximal element, then there exists a totally ordered subset of S which has no upper bound." but this is not helping me in my proof.
So the question is: what valid arguments (if any) can be used to demonstrate that a certain poset has no maximal element? There is another well-known "matematician tool" - that can lead to the conclusion I'm searching for?
 A: 
So the question is: what valid arguments (if any) can be used to demonstrate that a certain poset has no maximal element? There is another well-known "matematician tool" - that can lead to the conclusion I'm searching for?

It's impossible to say in generality what form the argument would take, but I think often one would be able to use proof by contradiction (suppose your poset has a maximal element, then show why a contradiction follows.) As a simplified example "Suppose $\mathbb N$ has a maximal element $n$. But then there's $n+1$ which is greater than $n$, oops!"

I need to prove that a certain partially ordered set S, which contains infinite (countable) elements, has no maximal element.

Zorn's lemma doesn't really help you at all with respect to this task.  From the statement you have something that concludes there is a maximal element, and from the contrapositive you conclude there must be an infinite ascending chain.   But what you want is something that concludes "there is not a maximal element." So, an argument other than what makes Zorn's Lemma work is called for.
I think if you need more advice you'll have to share what poset you're thinking of.
