An application to Zorn's lemma

Question

Let $S$ be a subset of $A_+$ the positive elements of a $C^\ast$-algebra $A$ which is weakly compact. I want to show that $S$ has a minimal and maximal elements.

I know that $S$ has a partial order since its a subset of $A_+$ we say $a\leq b$ for $a,b\in S$ if $b-a$ is a positive element. How can I show that any chain in $S$ has an upper bound?

Answer 1

Since $S$ is weakly compact, given a chain $\{a_j\}\subset S$, there is a weakly convergent subnet. Let $a\in S$ be the limit of the subnet. Is is then easy to see that $a$ is an upper bound for $\{a_j\}$.