Prove there is no contraction mapping from compact metric space onto itself This question is from Foundations of mathematical analysis by Richard Johnsonbaugh

The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a fixed point - someone look into this
How should one go about dealing with this question?

HINT: Let $p$ be the fixed point guaranteed by the earlier question. Show that there is an $x\in X$ that maximizes $d(p,x)$. Then show that $x\notin f[X]$.
