I am stuck on the following problem:
Prove that if $(X, d)$ is a complete metric space and $f : X\rightarrow X$ is a function with the property that there is a number $A < 1$ such that $d(f(x),f(y))\leq Ad(x,y)$ for all $x$ and $y$ in $X$ then $f$ has a unique fixed point.
Since $(X,d)$ is a complete metric space then every Cauchy sequence in $X$ converges.
I can pick $N$ such that for all $n,m > N$ then $d(x_n,x_m) < E$ for all $E > 0$.
To solve this I need to show that there exists an $x^*$ in $X$ such that $f(x^*) = x^*$. Any hints would be great!