Understanding Zorn's lemma. A lot of authors assume Zorn's lemma. I am told it is not an obvious mathematical fact, but I am having problems understanding why that is. 
Zorn's lemma states that if every chain in a partially ordered set $P$ has an upper bound in $P$, then it has a maximal element. 
Both the maximal element and the upper bound of every chain have to belong to $P$. 
Say any one chain of $P$ has an upper bound in $P$. Won't that be the maximal element by default. Is there some fine difference between maximal element and upper bound which I am unable to grasp?
Thanks in advance!
 A: The reason this is not "obvious" and instead it is a rather deep mathematical truth is that partial orders, even ones where every chain has an upper bound, can get pretty hairy and intangible, mathematically speaking.
When we think, and try to visualize structure in our heads, and a partial order is structure, we are limited by our imagination, which is often good at modeling finite objects, sometimes large enough that we think of them as infinite. But the mathematical truth is different.
Just as well, it is not obvious that every partial order has a maximal chain. If it was obvious, then indeed Zorn's lemma was obvious. But the fact remains that there are partial orders in the universe of sets which are wild beyond your imagination -- and then some.
Now, to your question, the assumption just says that a chain has an upper bound, which is an element of the partial order which is larger than the members of the chain. A maximal element, however, is an element which no point is strictly above it. Consider the interval $[-1,1]$ in the real numbers, the set of negative numbers has an upper bound, $0$. But this is not a maximal element. 
So Zorn's lemma tells us that if every chain has some upper bound, then there is a maximal element. And the chains will gradually become longer, stranger, and at some point they will exceed your imagination.
A: As others have pointed out, your statement of Zorn's lemma is ambiguous, and the one reasonable interpretation in English is not what is wanted (and in fact makes the statement false). This might be the source of your confusion. However, to address your questions at face value:
A maximal element of $P$ is an element such that no other is greater than it. That is, $x$ is maximal $p\not\geq x$ for any $p\in P$. Of course, because $P$ is not necessarily totally ordered, this is weaker than the notion of a maximum element, which demands that every other element is less than it.
Upper bounds require a bit more subtlety to define: Given a partially ordered set $P$ and a subset $S\subseteq P$, we say that $x$ is an upper bound of $S$ if every element of $S$ is less than $x$. Therefore, an upper bound is more similar to a maximum element than a maximal element. But it is importantly different because the upper bound does not need to belong to $S$ itself, just to $P$. 
(A classic example for when this distinction is important is that open sets on the real line never have maximum elements, but they could have upper bounds.)
