For the function has an absolute maximum at so, if then, is decreasing. For and we have .
But so, in uniformly .
Try the other part.
(a) Fn(x) = nxe^(-nx), x>= 0. Prove that the convergence is uniform for x>= @, where @ is any positive number. Why is the convergence not uniform on the interval 0<= x<= @??
(b) Fn(x) = (tan^-1)nx, all x. What conditions must a and b satisfy if the convergence is to be uniform on a <= x <= b ?