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$\{X_n\}_{n \geq 1}$ are i.i.d. random variables with standard Gaussian distribution. How do we prove the following:

$\mathbb{P}(-\infty$ = lim inf ${n \to \infty}$ $X_n$ $<$ lim sup ${n \to \infty}$ $X_n$ = $\infty )$ = 1

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closed as off-topic by Davide Giraudo, Dando18, Xander Henderson, Jyrki Lahtonen May 4 '18 at 18:06

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