Let $T : \Bbb R^n \to \Bbb R$ be a linear map. Let $A \subset \Bbb R^n$ be a convex, closed and bounded set. We know that there exists $x_0 \in A$ such that $\sup T(x) = T(x_0)$. Check that $\tilde x_0$ can be taken to be an extremal point of $A$.
I feel like I need to use the extremal point definition but do not know how to incorporate it.
Def.: $P$ is an extreme point of $S$ if there do not exist points $Q_1$, $Q_2$ of $S$ with distinct $Q_1$, $Q_2$ such that $P$ can be written in the form $P = t Q_1 + (1 - t)Q_2$ with $0 < t < 1$.
It is well know that $A$ is the convex hull of the set of extreme points. Hence $x_0= \sum\limits_{k=1}^{n} t_ix_i$ with $t_i >0,\sum t_i=1$ and each $x_i$ an extreme point of $A$. If $T$ does not attain its maximum value at any of the $x_i$'s then $T(x_0)=\sum\limits_{k=1}^{n} t_i T(x_i)$ which is strictly less than the maximum value of $T$ on $A$. This is a contradiction.