Well, arctangent is the inverse to tangent: y= arctan(1/x^2) is the same as 1/x^2= tan(y). I assume you know that as x goes to infinity, 1/x^2 goes to 0. So what must tan(y) be? And then, what must y be?
Now that's "intuitive" and not a "valid proof" but it is thinking like that that will lead you to a "valid proof". We want to show that in order to make arctan(1/x^2) as close to 0 as we please, , which is , , , .
So take and work backwards: if then .