I tried to do this
let h(x) = f(x) - g(x)
for every x in [0,1] h(x)≠0
then for every x in [0,1] h(x) > 0 ( or < 0)
but after this i could't find a contradiction
Since you need this by Monday morning, I can only assume it's either an exam question or homework for credit. So here are some hints:
Let a be a fixed point of f (f(a)=a) -- you need to prove why a exists. Now consider the sequence x1 = a and xn+1 = g(xn).
By the way, the above leads to a direct proof, not a proof by contradiction.
To start you off, suppose without loss of generality
You'll get a contradiction along the way.
Note: Using the auxiliary function h you defined, could be used for a direct proof, but at that point you guys are assumed to know Intermediate Value Theorem, etc.
Disclaimer: If my line of thought is wrong or it has missing holes, please correct and/or inform me.