Hello MHF, please help on this problem; (and by the way, we only did the continuity lesson )

We are given :

$\displaystyle f:[0,1]\rightarrow [0,1]$

$\displaystyle g:[0,1]\rightarrow [0,1]$

both$\displaystyle f$ and $\displaystyle g$ are continuious on$\displaystyle [0,1]$

$\displaystyle f(g(x)) = g(f(x))$

If $\displaystyle f(a) = a $ then $\displaystyle f(g(a)) = g(a)$

Show that $\displaystyle \exists \alpha \in [0,1] f(\alpha) = g(\alpha)$

P.S : use proof by contradiction