You only have to show that is bounded independently of and . Write , and is bounded by .
Qn: Show that the function f(x) = 1/(1+x^2) is uniformly continuous on R.
I tried to show that the function satifies the Lipschitz condition
|1/(1+x^2) - 1/(1+y^2)| = |(x^2 - y^2)/(1+x^2)(1+y^2)| = |(x+y)/(1+x^2)(1+y^2)||x-y| --- (*)
Then I observe that since (x-1)^2 >= 0. Hence 1+x^2 >= 2x and 1+y^2 >= 2y.
Thus (*) <= |(x+y)/4xy||x-y| = 1/4 |1/x + 1/y||x-y|
However, I am unable to proceed after that. How should i do it? I have also tried to prove it by using the epsilon-delta definition of uniform continuity but the argument is very similar to the one above.