When seen for the first time it can be difficult to immediately grasp the meaning of ⋂∞i=1⋃∞k=iAk.
I found that when I was a student and encountering this for the first time. This is one reason I included #7 on Examples Sheet 1. My advice is to pick it apart slowly.
Ak is the event that the kth toss is a head.
First: The event ⋃∞k=iAk is true (occurs) if and only if there is at least one head somewhere amongst the tosses i,i+1,i+2,… .
Second: When we put ⋂∞i=1 in front we create an event that is true if and only if the above is true for all i. So how can the result be true? Well, if and only if there are an infinite number of heads. (There might also be an infinite number of tails.)
In general, ⋂∞i=1⋃∞k=iEk is occurs (or is true) if and only if an infinite number of the events E1,E2,… occur.
The idea is similar to the "lim sup" of a sequence {a1,a2,…}, which is defined as
limi→∞supk≥iak.
How about ⋃∞i=1⋂∞k=iAk? That is the event that eventually we see only heads. Now perhaps you can see that
⋃∞i=1⋂∞k=iEk⊆⋂∞i=1⋃∞k=iEk.
Here is an article on it that might also help.
I found that when I was a student and encountering this for the first time. This is one reason I included #7 on Examples Sheet 1. My advice is to pick it apart slowly.
Ak is the event that the kth toss is a head.
First: The event ⋃∞k=iAk is true (occurs) if and only if there is at least one head somewhere amongst the tosses i,i+1,i+2,… .
Second: When we put ⋂∞i=1 in front we create an event that is true if and only if the above is true for all i. So how can the result be true? Well, if and only if there are an infinite number of heads. (There might also be an infinite number of tails.)
In general, ⋂∞i=1⋃∞k=iEk is occurs (or is true) if and only if an infinite number of the events E1,E2,… occur.
The idea is similar to the "lim sup" of a sequence {a1,a2,…}, which is defined as
limi→∞supk≥iak.
How about ⋃∞i=1⋂∞k=iAk? That is the event that eventually we see only heads. Now perhaps you can see that
⋃∞i=1⋂∞k=iEk⊆⋂∞i=1⋃∞k=iEk.
Here is an article on it that might also help.