Let $X1, ..., Xn$ be independent Bernoulli random variables, whose sum is $S_n$, each having probability $p > 1/2$ of being equal to $1$. Using Chernoff bound, show: $$\Pr\left[\lim_{n \to \infty} \frac{S_n}{n}>{1 \over 2}\right]\stackrel{(1)}{=} 1$$
My solution: For a finite $n$, Chernoff bound yields: $$\Pr[S_n>{n \over 2}]\geq 1-e^{-{\frac {1}{2p}}n\left(p-{\frac {1}{2}}\right)^{2}}$$ Therefore: $$\lim_{n \to \infty}\Pr[S_n>{n \over 2}]\stackrel{(2)}{=}1$$ How can I show that $(2)$ yields $(1)$?