I used proof by contradiction.
Let phi: A->A be a contraction with constant k<1
Assume that there exists 2 fixed points
phi(x1) = x1
phi(x2) = x2
According to the definition
|phi(x1) - phi(x2)|<=k|x1-x2|
as phi(x1) = x1 and phi (x2) = x2 then
|x1-x2|<=k|x1-x2|
which gives
1<= k
which contradicts the original condition that k<1....
Am i fumbling along the right lines, or is it all rubbish?