How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point?
What I know:
I know that all contractions are continuous and that completeness of $A$ means that all Cauchy sequences converge in $A$, but what can I do with these facts? And I think I also have to prove having two fixed points implies they are the same point to prove uniqueness. How would I do this?
HINT:
find $d(a_m,a_n)$ with $m>n$ and use the fact the series $1+k+k^2...$ is convergent hence the tail goes to $0$
now if it has $2$ fixed points say $x,y$ find relation between $d(x,y)$ and $d(f(x),f(y))$ where $f$ is the contraction