The law of the iterated logarithm

A function $\phi (x)$ is said to belong to the 'upper class' if, in the notation of this section, $\mathbb {P}\left(S_{n} > \phi (n) \sqrt{n} \text{ i.o.}\right) = 0$. A consequence of the law of the iterated logarithm is that $\sqrt{\alpha \log \log x}$ is in the upper class for all $\alpha > 2$. Use the first Borel-Cantelli lemma to prove the much weaker fact that $\phi (x) = \sqrt{\alpha \log x}$ is in the upper class for all $\alpha > 2$, in the special case when the $X_{i}$ are independent $N(0,1)$ variables.