Theorem 6.4.6: Max-Min theorem for Continuous Functions
In particular, f must be bounded on the compact set K.
With the work we have done previously, this proof is easy: Since K is compact and f a continuous function, f(K) is compact also. The compact set f(K) is bounded, so that f is bounded on K. The compact set f(K) also contains its infimum and supremum, so that f has an absolute minimum and maximum on K.
That's all !