Supremum of a sequence of i.i.d. standard normal random variables

Question

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. standard normal distributed random variables. Show, using Borel-Cantelli's Lemma, that $$\sup_{n\in \mathbb{N}} |X_n|= \infty \quad\mathbb{P}\text{-a.s.}$$

Anyone knowing how to do it? Thank you!

Answer 1

For any $M\in\mathbb{R}^+$ the probability that $(X_1,X_2,\ldots,X_N)$ belongs to $[-M,M]^N $ is given by $c^N$, where: $$ c = \frac{1}{\sqrt{2\pi}}\int_{-M}^{M}e^{-x^2/2}\,dx < 1-\frac{1}{2M}e^{-M^2/2}<1, $$ so the probability that every element of the sequence $\{X_n\}_{n\in\mathbb{N}}$ belongs to $[-M,M]$ is zero.