O.K showcase ,apart from your proof that uses the IVT AND which is correct ,the rest is not very clear .
So to avoid mixing up threads,let us start here by taking the problem in steps:
First we define the set , .
THEN we prove that is non empty and bounded from above and using the axiom of completeness we conclude that it has a supremum ,b.
THEN we claim that and to prove that we assume 1st and then .
THE question now is how do we prove that the above two cases are not true and we are forced to accept that:
??