How is this an application of the independence property of events? I'm currently working my way through Klenke's book on probability theory and do not understand a step in his proof of the Borel-Cantelli lemma (Theorem 2.7): The assertion is that for an independent family of events $(A_n)_{n\in\mathbb N}$ with $\sum_{n\in\mathbb N} \mathbb P(A_n) = \infty$ we have $\mathbb P(A^*) = 1$ where $A^* := \limsup_{n\in\mathbb N} A_n$. 
In his short proof he argues that for every $m\in\mathbb N$ $$ \mathbb P\left[\bigcap_{n=m}^\infty A_n^c\right] = \prod_{n=m}^\infty \mathbb (1-\mathbb P(A_n))$$
If this equation holds, the rest of the proof is clear to me. Independence, as he defines it, is that a family $(B_i)_{i\in I}$ of events is independent if for every finite $J\subseteq I$ the equation 
$$ \mathbb P\left(\bigcap_{j\in J} B_j\right) = \prod_{j\in J} \mathbb P(B_j)$$
holds true. And he's already proven that one can exchange events with their complements and the independence property is preserved. However, even if $(A_n^c)_{n\in\mathbb N}$ is known to be independent, then the above equation uses an infinite subset of the index set. So this equation cannot directly be obtained by independence? Where is the clue here?
 A: I write down the answer: We want to prove the following 
Claim: Let $(\Omega,\mathscr A,\mathbb P)$ be a probability space, $(A_n)_{n\in\mathbb N}$ a family of independent events. Then for every $m\in\mathbb N$ the equation
$$ \mathbb P\left(\bigcap_{n=m}^\infty A_n^c\right) = \prod_{n=m}^\infty \mathbb (1-P(A_n))$$
holds true.
Proof: Since $(A_n)_{n\in\mathbb N}$ are independent, so are $(A_n^c)_{n\in\mathbb N}$. Let $m\in\mathbb N$. By independence we get for every $N\geq m$:
$$\tag{$\star$}\mathbb P\left( \bigcap_{n=m}^N A^c_n\right) = \prod_{n=m}^N \mathbb P(A_n^c) = \prod_{n=m}^N (1-\mathbb P(A_n))$$
As $\mathbb P$ is a probability measure it is finite and continuous from above. Hence we get 
$$ \mathbb P\left( \bigcap_{n=m}^\infty A_n^c \right) = \lim_{N\to\infty} \mathbb P\left(\bigcap_{n=m}^N A_n^c \right) \overset{(\star)}{=} \lim_{N\to\infty} \prod_{n=m}^N (1-\mathbb P(A_n)) = \prod_{n=m}^\infty (1- \mathbb P(A_n)) $$
where the first equation holds by continuity from above, the second by independence and the third by properties of the real numbers. Note that the right hand side of the last equation is well-defined since every factor is bounded from below by $0$ and from above by $1$. Hence the sequence $\left(\prod_{n=m}^N (1-\mathbb P(A_n))\right)_{N\in\mathbb N}$ is decreasing and bounded from below by $0$ and therefore convergent. Its limit is written $\prod_{n=m}^\infty (1-\mathbb P(A_n))$. $\square$
