"Let f:[a,b]->[a,b] be a continuous function. Prove that there exists some point x in [a,b] such that f(x)=x."
Is it possible to do this WITHOUT using the intermediate value theorem?
This is a good problem. I would go about it using the squeeze theorem. Maybe start that for some x between a and b, f(a)<f(x)<f(b). Post back with any thoughts.