Let A_n denote the event that the event A occurs on the n…

Let $A_n$ denote the event that the event $A$ occurs on the $n$-th trial, and let $B_{n,m}$ denote the event that in the first $n$ trials the event $A$ occurs exactly $m$ times.
$1^{\circ}$ Express $B_{4,2}$ in terms of the $A_i$.
$2^{\circ}$ What is the meaning of the event $\displaystyle B_m=\bigcap_{n=1}^{\infty}\Big(\bigcup_{k=0}^{m}B_{n,k}\Big)$?
$3^{\circ}$ Are the relations $\displaystyle\bigcap_{n=1}^{\infty}A_n\subseteq\overline{B}$ and $\displaystyle\bigcap_{n=1}^{\infty}\overline{A_n}\subseteq B$ correct, where $B=\bigcup_{m}B_m$?

Step 1: Part $1^{\circ}$ — $B_{4,2}$

"Exactly two successes in four trials": choose which two trials are successful ($A_i$) and require the other two to fail ($\overline{A_i}$). There are $\binom{4}{2}=6$ choices:

The six cases are mutually exclusive (they differ in at least one trial), and $B_{4,2}$ is their sum:

\[ \boxed{B_{4,2} = A_1A_2\overline{A_3}\,\overline{A_4} + A_1\overline{A_2}A_3\overline{A_4} + A_1\overline{A_2}\,\overline{A_3}A_4 + \overline{A_1}A_2A_3\overline{A_4} + \overline{A_1}A_2\overline{A_3}A_4 + \overline{A_1}\,\overline{A_2}A_3A_4.} \]

Step 2: Part $2^{\circ}$ — the inner union

For a fixed $n$, the events $B_{n,0}, B_{n,1},\dots,B_{n,m}$ mean "$0$, $1$, …, $m$ successes in the first $n$ trials". Their union is

\[ \bigcup_{k=0}^{m}B_{n,k} = \text{"at most } m \text{ successes in the first } n \text{ trials"}. \]

Step 3: Part $2^{\circ}$ — the outer intersection

The intersection over $n=1,2,3,\dots$ requires this for every $n$: after 1 trial at most $m$ successes, after 2 trials at most $m$, …, after any number of trials at most $m$.

• If $A$ occurs at most $m$ times in the whole infinite sequence, then of course every initial segment has at most $m$ successes, so $B_m$ occurs.
• If $A$ occurs more than $m$ times, then after the $(m+1)$-th success, say at trial $N$, the first $N$ trials contain $m+1$ successes, so the condition fails for $n=N$.

\[ \boxed{B_m = \text{"in the whole infinite sequence of trials, } A \text{ occurs at most } m \text{ times"}.} \]

Note that $B_0\subseteq B_1\subseteq B_2\subseteq\cdots$ (at most $m$ implies at most $m+1$).

Step 4: Part $3^{\circ}$ — the event $B$

$B=\bigcup_m B_m$ means "for some number $m$, $A$ occurs at most $m$ times", i.e.

\[ B = \text{"}A\text{ occurs only finitely many times"}, \qquad \overline{B} = \text{"}A\text{ occurs infinitely many times"}. \]

(The index $m$ runs over $0,1,2,\dots$; whether it starts at $0$ or $1$ does not matter, because $B_0\subseteq B_1$.)

Step 5: Part $3^{\circ}$ — checking the inclusions

• $\bigcap_{n}A_n$ = "$A$ occurs in every trial". Then $A$ occurs infinitely many times, so we are in $\overline{B}$: \[ \bigcap_{n=1}^{\infty}A_n\subseteq\overline{B} \quad \textbf{— correct.} \]
• $\bigcap_{n}\overline{A_n}$ = "$A$ never occurs". Then $A$ occurs $0$ times, a finite number; in fact this event is exactly $B_0$: \[ \bigcap_{n=1}^{\infty}\overline{A_n} = B_0\subseteq B \quad \textbf{— correct.} \]

Both inclusions are strict: $\overline{B}$ also contains sequences like "success, failure, success, failure, …", and $B$ contains sequences with, say, exactly three successes.