Question

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.

Thank you.

Answer 1

Consider the set $(0,1)$ in $\mathbb R$. It has a least upper bound, but no maximal element.