First off, this isn't homework, but I'm doing research into large cardinal stuff so I wanna understand these theorems. I'm given this information to work with:
a cardinal $\mu$ is measurable if there exists a nonprincipal $\mu$-ultrafilter on $\mu$, that is, a nonprincipal ultrafilter that is closed under the intersection of $<\mu$-many sets in the filter.
a cardinal $\kappa$ is called weakly-compact if, for every complete total order $(C,<)$, if $|(C,<)|\ge\kappa$, then there exists a $B\subseteq C$ such that $|B|=\kappa$ and either $(B,\le)$ or $(B,\ge)$ is a well-order.
I am also given (and understand the proofs of) the facts that a topological space $X$ is compact iff every ultrafilter on $X$ converges, and that a total order $(X,<)$ is compact under the order topology iff $(X,<)$ is a complete order.
The two ways I see to do this are: 1. for a direct proof, constructing a well-order $(B,\le)$ or $(B,\ge)$ with $|B|=\mu$ from a non-principal $\mu$-ultrafilter, or for the contrapositive, 2. showing that if every $B\subseteq C$ is not a well-order under $\le$ nor $\ge$, then any $\mu$-ultrafilter on $\mu$ is principal. Not entirely sure how to show either of these, however. Any tips for how to approach this?
I do know that given a $\mu$-ultrafilter $\mathcal{F}$ and an injection $f:\mu\to C$, the set $\{f(A):A\in\mathcal{F}\}$ is a $\mu$-ultrafilter (I have verified this), so perhaps I could work with the $\mu$-ultrafilter ordered under inclusion, but again, not entirely sure until I try it.