Letf be a continuous function on the closed interval [0; 1], and suppose that f(0) < 0

and that f(1) > 0. By considering the set S= {x E [0,1] :f(x)_<0) prove that there exists

a x' such that f(x')=0

S is non empty as x=0 is a member and bounded above by 1, so by the completness axiom Sup(S) exists. Sup(S)_> x E S and Sup(s) _< 1. Not sure where to go from here. Thanks in advance.