Let Y_1, \dots, Y_n, \dots be random variables with characteristic functions \varphi_1, \dots, \varphi_n, \dots. Prove that if \exists_{\delta > 0} \lim\limits_{n \to \infty} \varphi_n (t) = 1 for |t| < \delta then Y_n \stackrel{d}{\to} 0.

So I have to show that \lim\limits_{n \to \infty} F_n (t) = F(t), where F_1, \dots, F_n, \dots are distribution functions for Y_1, \dots\ Y_n,\dots and F(t) = I_{[0, \infty)}(t).

Could someone give me a hint?