Try experimenting with the new function g(x) = f(x) - x, and think about the Intermediate Value Theorem.
I'm trying to prove that if f : [a,b] ---> [a,b] is continuos, there exists an x in [a,b] such that f(x) = x
I've formed a sort of geometric proof - if you take the square [a,b] and draw the diagonal down the middle representing the identity function, then any function defined on the interval [a,b] has to intersect the line at least once.
I'm having trouble formalizing it though. Help would be greatly appreciated...