Fair coin toss probability asymptotics [duplicate]

Possible Duplicate:
Asymptotics for a partial sum of binomial coefficients

A fair coin is tossed $n$ times, let $A_n$ be the number of heads and $B_n$ the number of tails and $C_n = A_n - B_n$.

Prove that for every $0<a<1$ holds

$\left(\mathbb{P}\left(C_n \geq an \right)\right)^{1/n} \to \frac{1}{\sqrt{(1+a)^{1+a}(1-a)^{1-a}}}$

as $n \to \infty$.

I would be grateful for any ideas or hints. Thanks.