zorn's lemma

Let \(P\) be a poset. If every chain \(S\) has an upper bound in \(P\), then \(P\) has at least one maximal element.

Question for me: What is an example of \(P\) that has a chain \(S\) with upper bound not in \(P\)? To even be comparable with the elements of \(S\), doesn't the upper bound have to be \(P\)? Answer: Oh I guess you can define \(P\) to be anything you want. Two elements can be comparable, but you can draw the boundaries of \(P\) to exclude one of them.