Real Analysis problem Suppose that f: [a,b] -> [a,b] is continuous.

Prove that f has a fixed point. That is, prove that there exists c belonging to [a,b] such that f(c) = c.
Hint: consider g(x) = f(x) - x
Thanks!

By the hint,
g(a) = f(a) - a >= a - a = 0
g(b) = f(b) - b <= b - b = 0.

Then by the intermediate value theorem, there is some c in [a,b] such that
f(c) - c = g(c) = 0,
and therefore,
f(c) = c.