I think I kinda know the direction this proof needs to go but I'm not sure how to word it all. Any help would be appreciated.

Let $\displaystyle f:[0,1]\rightarrow [0,1]$ be continuous. Prove that there is a $\displaystyle c\in [0,1]$, such that $\displaystyle f(c)=c.$ [Hint: consider $\displaystyle g(x)=x-g(x)$

I'm thinking a proof by contradiction might be apprpriate here. Assume that there is no c that works and then find a contradiction by using the hint somehow????