Consider a series of $n$ i.i.d random variable $X_i, 1\leq i \leq n$ \begin{align*} X_i = \begin{cases} 1 &\quad w.p \ \ \ p_1\\ -1 &\quad w.p \ \ \ 1-p_1\\ \end{cases}\\ \end{align*} And another series of $n$ i.i.d random variable $Y_i$ \begin{align*} Y_i = \begin{cases} 1 &\quad w.p \ \ \ p_2\\ -1 &\quad w.p \ \ \ 1-p_2\\ \end{cases}\\ \end{align*}
Where $\ 0 < p_1 <p_2 <1/2 \ $. I want to find the following limit: \begin{align*} \lim_{n \to \infty} \frac{Pr(\sum_{i=1}^n X_i \ge0)}{Pr(\sum_{i=1}^n Y_i \ge0)} \end{align*}
I tried using CLT and Lhopital Theorems by assuming that both denominator and numerator are gaussian at the same time and got that the limit is zero. My question is how to do this without this assumption? Alternatively, is there a better way to show the limit is zero?