I am confused by the Zonrn's lemma and Least Upper Bound axiom:
(1) Least upper bound axiom: every subset of real number if has an upper bound then has a least upper bound.
(2) Zorn's lemma: Let (A, <=) be a partially ordered set. If every chain in A has an upper bound then A has maximal.
I think if each chain in A has an upper bound then the chain should have maximal (as the Least upper bound axiom state above) hence the set of maximal of A should be the set of maximal of chains of A.
My intuition want to believe (though I know it is wrong) that Zorn's lemma should be merged with Least Upper bound axiom into the form:
Let (A, <=) be a partially ordered set. If a chain in A has an upper bound then it has maximal and hence A has maximal.
Could you give me an encounter-example to show that if a chain has an upper bound then it probably has no maximal.