Use proposition 2.3 to provide an alternative proof of theorem 2.2 Theorem 2.2 A continuous functional on

Question:

Use proposition 2.3 to provide an alternative proof of theorem 2.2
Theorem 2.2
A continuous functional on a compact set achieves a maximum and a minimum.
Proof Let M = supx∊X f (x). There exists a sequence xn in X with f (xn) → M. Since X is compact, there exists a convergent subsequence xm → x* and f (xm) → M. However, since f is continuous, f (xm) → f (x*). We conclude that f (x*) = M.