I have an issue understanding how the Central Limit Theorem and the Strong Law of Large Numbers coexist. I have constructed the following example, but I cannot find the mistake I made and am therefore asking for your help. Thank you in advance!
Let $X_1, X_2, \dots$ be i. i. d. random variables on $(\mathbb{R}, \mathbb{B}(\mathbb{R}))$ and $X_1 \in L^2$. Then the Strong Law of Large Numbers tells us that for $S_n = \sum_{k = 1}^{n} X_k$ we have $\frac{S_n}{n} \rightarrow \mathbb{E} [X_1]$ almost surely.
But if we consider $P(\frac{S_n}{n} > \mathbb{E}[X_1]) = P(\frac{S_n - n \mathbb{E}[X_1]}{n}>0) = P(\frac{S_n - n \mathbb{E}[X_1]}{\sqrt{nVar(X_1)}} > 0)$ and take the limit we get by the CLT that $lim_{n \rightarrow \infty} P(\frac{S_n - n \mathbb{E}[X_1]}{\sqrt{nVar(X_1)}} > 0) = P(N > 0) = \frac{1}{2}$.
However, since $\frac{S_n}{n}$ converges almost surely to $\mathbb{E}[X_1]$, we have that $lim_{n \rightarrow \infty} P(\frac{S_n}{n} = \mathbb{E}[X_1]) = 1$. Contradicting that the probability above equals $\frac{1}{2}$ since it should be $0$.
Where did I make a mistake?
Also please note that I implicitly assumed that the variance is larger than $0$.