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

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

Could someone give me a hint?