Moving a limit inside integral and uniform convergence.

Question

I want to prove $$ \lim_{\alpha\to0^+} \int_0^\infty \frac{\sin^2x}{x^2+\alpha^2}=\int_0^\infty \frac{\sin^2x}{x^2}.$$ It seems trivial but I know that moving the limit inside the integral is not always allowed, but it should be allowed if the sequence of functions converge uniformly, if I am not mistaken. If $$f_n=\frac{\sin^2x}{x^2+n^2},$$ then to show show unifrom convergence: $$\lim_{n\to\infty}\lVert f_n-f \rVert=0.$$ I get $$\lim_{n\to\infty}\lVert \frac{\sin^2x}{x^2+n^2} -0\rVert=\lim_{n\to\infty}\sup\rvert \frac{\sin^2x}{x^2+n^2}\lvert,$$ and this is where I am stuck. How do I find $$\sup\rvert \frac{\sin^2x}{x^2+n^2}\lvert?$$