Prove that the sequence $X_n = Z_n/ \sqrt{1+\log{n}}, ~ n\geq 1$ is almost surely bounded for $Z_n ~ i.i.d. ~ N(0,1)$. Let $Z_1,Z_2,\ldots$ are i.i.d. $N(0,1)$ random variables. Let $X_n = Z_n/ \sqrt{1+\log{n}}, ~ n\geq 1$. Question is to show that $X_n$ is almost surely bounded. 
Proceeding as following :
Step 1. I have a proof that $P(|X_n|\geq \lambda)\leq n^{-\lambda^2/2} \exp(-\lambda^2/2)$.
Step 2. Then how do I bound $P(sup_{n\geq 1} |X_n| > \lambda)$ for $\lambda \geq 2$, so that I can show that $X_n$ is almost surely bounded?

Hint: $\sup |X_n|$ is greater than $\lambda$ if and only if at least one of the $|X_n|>\lambda$. So you need a bound for the probability of the union:
$$
\mathbb P\left(\bigcup_{n\ge0} |X_n|>\lambda\right)\le \dots
$$
What ways do you know to bound the probability of the union of events?
