Proof for Statement regarding Borel Cantelli Lemma

Question

I am struggeling to apply the Borel-Cantelli lamma to the following problem:

Let $ A, A_1, A_2,\dots \in F $ in the probability space $ (\Omega, F, \textit{P}$) with $ \sum_{n \in \mathbb{N}} P ( A_n ∩ A) < ∞ $.

Now i need to show that

$P(\limsup_{n \to \infty} A_n) \leq 1 - P(A) $

Due to the additional information of the convergence of the sum, I am quite sure that the Borel-Cantelli - Lemma is needed, which says that $ P(\limsup_{n \to \infty} A_n ∩ A) = 0 $. How am I get the A into the original equation? Thanks in advance

Answer 1

$\newcommand{\pr}{\operatorname{Pr}}$The regular Borel-Cantelli lemma shows that the event: $$B=\bigcap_{n\ge1}\bigcup_{k\ge n}(A_k\cap A)=A\cap\bigcap_{n\ge1}\bigcup_{k\ge n}A_k=A\cap\limsup_{n\to\infty}A_n$$Has probability zero. You want to show that $\pr(\limsup_{n\to\infty}A_n)\le1-\pr(A)$.

Consider that: $$\pr(A\cup\limsup_{n\to\infty}A_n)=\pr(A)+\pr(\limsup_{n\to\infty}A_n)-\pr(B)$$

Can you finish?