Is my proof of convergence for a sum of random variables correct?

Question

Let $ X_i \text{ for } i \in \mathbb N $ be a sequence of independent and identically distributed random variables, with $X \sim \mathscr N_{0,1}$ and $ \alpha>2 $.
$ S_n := \sum_{i=1}^n X_i$ and $A_n :=\{ S_n \gt \sqrt{\alpha n \log(n)} \}$. Prove that: $$ \mathbb {P} \left( \limsup_{n\to \infty} \sum_{i=1}^n X_i > \sqrt{\alpha n \log(n)} \right) = 0 $$ My approach was the following:
I can use the following theorem: $$ X_i \text{ are independent and identically distributed random variables } \forall \epsilon \gt 0 \text{ and } \mathbb{V}(X_i) \lt \infty \text{ follows } \lim_{n \to \infty} \mathbb{P} \left( |\frac{1}{n} \sum_{i=1}^n X_i-\mathbb {E} (X_1)| \gt \epsilon \right) = 0 $$ We know that $X_i$ satisfy the conditions, meaning I can just set $ \epsilon $ to be $ \frac{\sqrt{\alpha n \log(n)}}{n} $ and be done right?
However, what about the $ \limsup $ ?