Boole's inequality, Big Union/Intersection Symbol and a Sequence of Biased Coin Flips I am confused with the notation used for an example using Boole's inequality.
First of all, here's the question:
"Consider a sequence of tosses of biased coins. Let $A_k$ be the event that the $k$th toss is a head. Suppose $P(A_k) = p_k$. The probability that an infinite number of heads occurs is 
$$ P(\bigcap_{i=1}^{\infty}\bigcup_{k=i}^\infty A_k) \leq P(\bigcup_{k=1}^\infty A_k) = p_i + p_{i+1} + ...$$
Hence if $\sum_{i=1}^\infty p_i < \infty$ the right hand side can be made arbitrarily close to 0.
This proves that the probability of seeing an infinite number of heads is 0.
If we assume coin tosses are independent then the reverse is also true: if $\sum_{i=1}^\infty p_i = \infty$ then P(# of heads is infinite) = 1"
As far as I can understand,
$$ \bigcap_{i=1}^{\infty}\bigcup_{k=i}^\infty A_k = (A_1 \cup A_2 \cup ... \cup A_\infty) \cap (A_2 \cup A_3 \cup ... \cup A_\infty) \cap .. \cap A_\infty = A_\infty$$
Which is just the event of a head for $k \rightarrow \infty$ - clearly I'm misunderstanding the notation in some way.
 A: There is no such event as $A_\infty$.  No matter how long you count, you never get to $\infty.$  Informally, $\bigcup_{k=i}^\infty A_k$ is the set of heads that occur on or after toss number $i$.  If the intersection of these is nonempty, it must be that heads occurs infinitely often.  If not, there would be some toss $n$ on which the last heads occurred, and then we'd have $\bigcup_{k=n+1}^\infty A_k=\emptyset$, and then the intersection would be empty, and have probability $0$.
A: In your problem, there is no notion of $A_\infty$. So the last thing you wrote does not make any sense. 
Now let us try to find the meaning of that strange intersection. Before that, set $B_i = \bigcup_{k=i}^\infty A_k$ and observe that $B_i$ is the event that you get at least one head on or after the $i$-th toss. Now, if you get finitely many heads, then at least one $B_i$ won't occur. In other words, for getting infinitely many heads, we need all the $B_i$'s to occur. Thus, the event of getting an infinite number of heads is $\bigcap_{i=1}^\infty B_i.$ 
Interestingly, here the sequence of events $B_i$ has the property that $B_{i+1}\subset B_i$ for every $i\ge 1.$ Hence we may denote $B_\infty = \lim_{n\to \infty} B_n$ and in that case you can show that $\bigcap_{i=1}^\infty B_i = B_\infty.$
