limit of generalized functions $f_{\epsilon}(x) = \frac{sin^2(x/ {\epsilon})}{\pi x^2}$

Question

I am trying to find this limit. Let's conisder $\phi \in C_0(R)$. $$\int^{\infty}_{-\infty} \frac{sin^2(x/ {\epsilon})}{\pi x^2}\phi(x)dx$$ = $$\int^{\infty}_{-\infty} \frac{sin^2(y)}{\pi \epsilon y^2}\phi(\epsilon y)dy=I_{\epsilon}$$. I know that ansewer is $\delta(x)$. Then we have to proof that $lim_{\epsilon \to 0} I_{\epsilon}=\phi(0)$. $$|I_{\epsilon}| \leq \int_{supp \phi(\epsilon y)} \frac{sin^2(y)}{\pi \epsilon y^2}|\phi(\epsilon y)|dy$$. But this $\epsilon$ ruined all my integral estimates, and it is going to infinity. Please, give me some hints.