Can you help me solving the following exercise? I should use the Borel-Cantelli Lemmas, but I don't know how.
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and let $(X_n)_{n \in \mathbb N}$ be a sequence of i.i.d. random variables having a continuous distribution function. Define the following two sequences of events for $n \geq 1$:
$$ A_n := \{X_n > X_i,\ \forall i \in \{0,...,n−1\}\}, \quad B_n = A_n \cap A_{n+1} $$
Compute $\mathbb P(A_n\ \text{i.o.})$ and $\mathbb P(B_n\ \text{i.o.})$. Justify your answer.
First it is not hard to see that $A_n$ are pairwise independent. Also, $\mathbb{P}(A_n)=\frac{1}{n+1}$ and $\mathbb{P}(B_n)=\frac{1}{(n+1)(n+2)}$.
Since $\sum \mathbb{P}(A_n) = +\infty$ and $A_n$ are pairwise independent, by Borel-Cantelli lemma $A_n$ happens infinitely often almost surely.
Also, since $\sum \mathbb{P}(B_n) < \infty$, again by Borel-Cantelli Lemma $B_n$ happens finitely often almost surely. Thus the first probability is $1$ while the latter is $0$.