Suppose that F is a function that is continuous at all points in the interval [0, 1]. Suppose also that for all b < 1 there is some x such that 0 x 1 and sin(F(x)) > b. Prove that there is some point z in the interval [0, 1] such that sin(F(z)) = 1. Your proof must quote all relevant theorems from the text and show that their hypotheses are satisfied.

reference to textbook - Calculus: One and several Variables, 10th edition