Help: an intersection of decreasing sequence example Help! I don't understand the following example:
A die is thrown infite times. Let be $A_n$ be the event corresponding to get the event $A=\{2,4,6\}$ on each one of the n-firsts experiments. Clearly, $A_n \supseteq A_{n+1}$ and $P(A_n)= \frac 1{2^n}$, so : 
$$\lim_{n\to \infty} A_n = \underset{n=1}{\overset{\infty} {\cap}} A_n $$
I don't understand why $A_n \supseteq A_{n+1}$ 
I'd say that $A_1 = \{A\}$, $A_2 = \{A,A\}$, $A_3 = \{A,A,A\}$, for $n=1,2,3,\dots$   so   $\bigcap_{n=1}^{\infty}A_n = A_1 $ and I think that this should be Ø...
 A: Suppose that you have $2$ on your first throw and $4$ on the second throw. So the event belongs to $A_2$. Since you have $2$ on the first throw, the event is also in $A_1$. A similar reasoning works for $A_n $.
Let $S=\{1,2,3,4,5,6\}$. Then, in your notation, $A_1=\{A,S,S,\cdots\},$ $A_2=\{A,A,S,\cdots\}$ etc. 
A: If I understand well then $A_n$ is meant to be the event that the first $n$ throws all result in an even outcome.
If we state that $E_k$ is the event that at the $k$-th throw the outcome is even then:$$A_n=\bigcap_{k=1}^n E_k$$
This confirms that $A_1\supseteq A_2\supseteq A_3\supseteq\cdots$
So $A_n$ is getting smaller and smaller and has a limit.
Also we have:$$\bigcap_{n=1}^{\infty}E_n=\bigcap_{n=1}^{\infty}A_n=\lim_{n\to\infty}A_n$$
A: Remember that events are defined as sets of outcomes, and each outcome is a full description of everything that happens in the experiment. In this example, each outcome is an infinite sequence $(x_1,x_2,x_3,\dots)$ of throw results.
For example, the event $A_2$ contains the outcomes $(2,4,1,3,3,3,\dots)$, $(6,6,6,6,6,6,\dots)$, and $(4,2,5,2,5,2,\dots)$, but not the outcome $(2,3,1,1,1,1,\dots)$. However, all four of these outcomes are elements of $A_1$. In fact, every sequence that starts with at least two even numbers also starts with at least one even number, so $A_2\subseteq A_1$.
You can think of "$A\subseteq B$" as meaning "if $A$, then $B$".
