how to show that $1/(1+x^2)$ is a contraction?

I am trying to show that $1/(1+x^2)$ is a contraction but I cannot find the contraction factor. So far I got: $\lvert \frac{1}{1+x^2}-\frac{1}{1+y^2}\rvert=\lvert\frac{x^2-y^2}{(1+x^2)(1+y^2)}\rvert=\lvert\frac{(x-y)(x+y)}{(1+x^2)(1+y^2)}\rvert$. So I got x-y in the numerator, but how do I find a bound and get rid of al the other terms?