Consider an LP with an unbounded feasible region, but whose objective is bounded. Maybe something like
$$\max x_1-x_2$$ $$\text{s.t. }x_1\le 1$$ $$x_1,x_2\ge 0$$
The proofs I've seen that at least one extreme point is optimal all assume that the feasible region is bounded, so that any point in the feasible region is a convex combination of extreme points. This isn't true here, though - is there an alternative way to prove this?