Find a closed interval topology

[g1990]

Homework Statement

Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property.

Homework Equations

X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other words, an upper bound where any number less that it is not an upper bound.
Compact means that every open cover has a finite subcover.

The Attempt at a Solution

Let A be a subset of X that is bounded from above. I know I should try to find a closed interval from this, but I'm not sure where to get it from.

 

[nonequilibrium]

Hello,
I'll try to give an intuitive explanation.

"Let A be a subset of X that is bounded from above."
If A has a greatest element, we're done. If A does not, we can focus on an interval B of the form [...[ with $$A \cap B = B$$. Note that B has no greatest element thus there is an open cover that has no finite subcover (is there?).
Define $$C = \overline{B}$$.

• If C = B, then B is closed and thus compact (CONTRADICTION! (?))
• So C has at least an element more than B. What is this element?