Zorn's Lemma and Posets If $A$ is a poset in which every chain has an upperbound in $A$ ($a$ be any element in $A$). There exists at least one maximal element $m$ in $A$ such that $m \geq a$.
Whats the difference between this and Zorn's Lemma?
 A: I suppose that you meant to say that for every $a\in A$ there is a maximal element larger than $a$ itself.
Zorn's lemma only guarantees us a maximal element. But perhaps there is only one? Perhaps some elements are not below any maximal element?
This slight variant of Zorn's lemma tells us that above any element there is a maximal one. This is a stronger statement, at first sight. Of course it proves Zorn's lemma, but as it turns out the opposite is true as well. 
Assume Zorn's lemma, and let $A$ is a partially ordered set in which every chain has an upper bound. Given $a\in A$ we define $C_a=\{x\in A\mid a\leq x\}$, and we can show that $C_a$ with the restricted order is also a partial order in which every chain is bounded. Zorn's lemma assures that there is $m\in C_a$ which is maximal, but it is not hard to verify that $m$ is maximal in $A$ as well, and therefore any $a\in A$ lies below a maximal element.
So while there is no actual difference, the statements are different. Whereas Zorn's lemma only assures us that some maximal element exists, this version assures us that every $a\in A$ lies below one.
