Question on Zorn's lemma Zorn's lemma states

Suppose a partially ordered set $P$ has the property that every chain (i.e. totally ordered subset) has an upper bound in $P$. Then the set $P$ contains at least one maximal element.

Now, of course, the rationals $\mathbb{Q}$ satisfy the condition that every chain has an upper bound in $\mathbb{Q}$. But $\mathbb{Q}$ does not contain a maximal element, a contradiction.
Where is my line of reasoning going wrong?

What is an upper bound of the chain $\mathbb{Z}$ in $\mathbb{Q}$?  (or $\mathbb{Q}$ itself, for that matter.)
