Let $S_n=X_1+X_2+X_3+...+X_n$ where $X_i$ are i.i.d with $E[X_i]=0$ and finite variance.
In a course I'm taking, the following solution was given:
$$\lim_{n \to \infty} P(S_n\geq 0)=1-\Phi(0)=0.5$$
While this makes some intuitive sense, I'm wondering where the mistake in the following line of reasoning using the law of large numbers is :
$$\lim_{n \to \infty} P(S_n\geq 0)= \lim_{n \to \infty} P(\frac{S_n}{n}\geq 0) \geq \lim_{n \to \infty} P(\frac{S_n}{n}= 0)=1 $$
The law of large numbers says that $P\big( \lim_{n \to \infty} \frac{S_n}{n}= 0 \big) = 1$, but that's not the same assertion as $\lim_{n \to \infty} P\big( \frac{S_n}{n}= 0 \big) = 1$.