Hello, I need to show that h(x) = 1/(x^2 +1) is uniformly continuous on R

This is what I have so far:

Let x,y be in R

I know |x-y|< delta

I want to show |f(x)-f(y)|< epsilon taking some delta

so, |1/(x^2 +1) - 1/(y^2 +1)|

= |(y^2-x^2)/(x^2 +1)(y^2 +1)|

= |(x-y)(x+y)/(x^2 +1)(y^2 +1)|

< |delta(x+y)/(x^2 +1)(y^2 +1)|

so i could chose an epsilon = (x^2 +1)(y^2 +1)/(x+y)

so that = epsilon.

However, to be uniformly continuous, my choice of delta shouldn't depend on x to be uniformly continuous.

please provide me with some feedback

thanks