In the proof to this theorem, how does taking the supremum of (8), yield the contradicting inequality? We know from the premises that f(x) < M, for all x in I. However, the statement f(x) <= M - (1/C) for all x in I does not in any way lead to the contradiction stated in the proof. Here is my reasoning:
f(x) < M for all x in I. f(x) <= M - (1/C). So, taking the second statement into consideration, it is less than or equal to, but the first condition holds in either less than or equal to.