This is NOT a true statement. The "intermediate value theorem" does NOT say "f(x)= x has to exist at least once".
You appear to be missing a statement. Did you intend this to be a "proof by contradiction" and say "suppose, for some x, [tex]f(x)\le x[tex]"?
BUT it is given thatf(x) x for every x.
Therefore f(x) > x , always.
Would that work as a "mathematical" proof for college etc..?