Exsistence of limit $\lim_{k\to \infty} \prod_{i=1}^{k}P(A_i)$ and $\lim_{k\to \infty}P(\bigcap_{i=1}^{k}A_i)$

Question

Let $(A_k)_{k\in\mathbb{N}}$ be a sequence of events. Argue that both limits exist $$\lim_{k\to \infty} \prod_{i=1}^{k}P(A_i)\quad \quad \lim_{k\to \infty}P(\bigcap_{i=1}^{k}A_i)$$

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I am not sure on how to answer this at all. I would have thought that they didn't as my book states the definition of independent probability only for finite values of $J\subseteq I$

$$P(\bigcap_{i \in J}A_i)=\prod_{i\in J}P(A_i)$$

Now the question above is considering cases where $I$ is the natural numbers and and the cardinality of $J$ is $\infty$.

How do I argue that these limits exists?

Answer 1

Let $\pi_{n}=\mathbb{P}(A_{1})\cdots\mathbb{P}(A_{n})$. Note that $\pi_{n}\geq0$ since it is a product of nonnegative numbers. Morever, $\pi_{n}\geq\pi_{n+1}$ since $\mathbb{P}(A_{n+1})$ is between zero and one. That is, $(\pi_{n})_{n}$ is a nonincreasing sequence bounded below by zero. By the monotone convergence theroem, this sequence converges.

Hint: Use a similar argument for the other sequence.

Hint 2: Let $A^{(k)}$ be the intersection of the first $k$ sets. Then, $A^{(n+1)} = A^{(n)} \cap A_{n + 1}$. What does this tell you about $\mathbb{P}(A^{(n+1)})$ and $\mathbb{P}(A^{(n)})$?