Can someone give me an idea how to generalize Banach's fixed-point theorem for complete metric spaces such that the constant contraction coefficient $c$ (as in $d(Tx,Ty)\leq c \ d(x,y)$ ) may be replaced by a sequence $(r_n)_{n \in \mathbb{N}}$ so that $T:X\rightarrow X$ has a unique fixed point if $d(Tx,Ty)\leq r_n d(x,y)$ holds for all $x,y \in X$ and $\sum r_n$ converges?
I tried modifying the original proof, but of $\sum r_n$ only converges (not necessarily with a limit in the interval $[0,1)$ I can't use the original idea...