Let $\{X_n\}$ be a sequence of independent Gaussian random variables with $\mathbb{E}\, X_n = 0$ for all $n \geq 1$. Find the probability of the event $$ \limsup_{n\to \infty} \big\{ X_n X_{n+1}> 0 \big\} $$
My first thought is that it should be 1 since Gaussians are always positive for a finite value. I was thinking of applying Borel-Cantelli and was trying something along the lines of \begin{align*} \mathbb{P} \big( \limsup_{n\to \infty} \big\{ X_n X_{n+1}> 0 \big\}\big) &= \mathbb{P}\big( X_n X_{n+1} > 0 \,\,\, i.o. \big) \\ &\leq \mathbb{P}\big( \big\{ X_n X_{n+1}> 0 \,\,\, i.o \big\} \cap \big\{ X_{n+1} > 0 \,\,\, i.o\big\} \big)\\ &= \mathbb{P}\big( \big\{ X_n X_{n+1}> 0 \,\,\, i.o \big\}\big) \,\,\mathbb{P}\big( \big\{ X_{n+1} > 0 \,\,\, i.o\big\} \big) \,\,\,\, \text{(by independence)} \end{align*} I'm not sure I'm thinking of this problem right, though.