Use of Zorn lemma in a proof of Kakutani theorem

Question

I'm trying to understand the first sentence in the proof of the Kakutani fixed point theorem here : https://mathweb.ucsd.edu/~nwallach/haarmeasure.pdf (page 2)

So let $K$ be compact in a locally convex topological vector space. I consider a set $\mathcal F$ of parts of $K$ such that every element of $\mathcal F$ is

• non empty
• compact
• convex

I suppose that $\mathcal F$ is totally ordered by the inclusion : if $A$ and $B$ belong to $\mathcal F$, then $A\subset B$ or $B\subset A$. (here I say $A\leq B$ when $B\subset A$ because I'm searching for a minimum, not a maximum)

Let $M=\bigcap_{A\in\mathcal{F}}A$.

I cannot prove that $M$ is non empty.

QUESTION: how to prove that $M$ is non empty ?

EDIT: full proof. Search for the label THOooWXQFooQrWcLY

Answer 1

Here is an answer using PhoemueX's comment.

Two preliminary results.

THEOREM (wikipedia) A space is compact if and only if every family of closed subsets having the finite intersection property has non-empty intersection.

LEMMA Every totally ordered finite subset of an ordered set has a minimum.

ANSWER TO THE QUESTION

The set $\mathcal{F}$ is totally ordered for the inclusion. So every finite part $\mathcal{F}_0$ has a minimum which is the intersection of every element of $\mathcal{F}_0$. Thus $\mathcal{F}$ has the finite intersection property.

Since $K$ is compact, and since the elements of $\mathcal{F}$ are closed (from compact), the intersection of $\mathcal{F}$ is non empty.