I started this year to study probability and I'm using A Probability Path of Resnick, I'm trying to solve this exercise but I can't see the link between the Borel-Cantelli's Lemma that I studied and the exercise. Can someone help me and give a Hint for the second question too?
On
I'll try to provide some minimal hints to get you un-stuck. There's still a fair bit to do after these hints; please let me know if you need more help.
(a) Consider the complementary events, $A_n^c$, for which $\mathbb P(A_n^c) \to 0$. This isn't enough to guarantee that $\sum \mathbb P(A_n^c) < \infty$, but you can argue that it has a subsequence that has this property. (How?)
(b) Let $\Omega = (0, 1)$, and let $$A_n = \{x \in (0,1) : \text{the binary representation of $x$ has a 1 in its $n^{\text{th}}$ position}\}.$$ Note that $\mathbb P(A_n) = 1/2$ for all $n$.
Since $\mathbb{P}(A_n)\xrightarrow{n\rightarrow\infty}1$, $\mathbb{P}(A^c_n)\xrightarrow{n\rightarrow\infty}0$.
Set $A_{n_1}=A_1$. Once $A_{n_1},\ldots,A_{n_k}$ ($n_1<\ldots<n_k$) have been selected, let $A_{n_{k+1}}$ so that $n_{k+1}>n_k$ and $\mathbb{P}(A^c_{n_{k+1}})\leq\frac12\mathbb{P}(A^c_{n_k})$. Then $$\sum_k\mathbb{P}(A^c_{n_k})<\infty$$ an so, $\mathbb{P}\Big(\bigcap_k\bigcup_{m\geq k}A^c_{n_m}\Big)=0$. Thus $\mathbb{P}\Big(\bigcup_k\bigcap_{m\geq k}A_{n_m}\Big)=1$ and (a) follows.