Theorem 6.4.8: Bolzano Theorem
With the work we have done so far this proof is easy. Since (a, b) is connected and f a continuous function, the interval f( (a,b) ) is also connected. Therefore that set must contain the interval (f(a), f(b)) (assuming that f(a) < f(b) ). Since f(a) and f(b) have opposite signs, this interval must include 0. Therefore, 0 is in the image of (a, b), or equivalently: there exists a c in open interval (a, b) such that f(c) = 0.
That's it !