Understanding continuity of probabilities I am reading a book called All of Statistics by Larry Wasserman that includes a theorem called "continuity of probabilities."

It says that if $A_{n} \rightarrow A$ then $P(A_{n}) \rightarrow P(A)$ as $n \rightarrow \infty$

I am trying to understand what that statement means in English and to understand its import. How can one probability imply another? 

I understand that $A_{n}$ can be broken into lots of disjoint $B_{n}$ which can be added up to form $A_{n}$ -- but I don't see the big picture or how that relates to sample spaces.
If we have $A_1 \subseteq A_2 \subseteq A_3 \subseteq \dots$ and $A=\cup_{n=1}^\infty A_n,$ we write $A_n \nearrow A.$   Given this assumption, a probability measure satisfies $\lim_{n\to \infty} P(A_n) = P(A).$  The word "continuity in probability" is analogous to continuity at point for a function (e.g.  As $x$ approaches $c$,  we have $f(x)$ approaches $f(c).$ )
I was trying to find the use of this theorem in a practical sense and came up with a following example.

Suppose you have some sort of distribution from $[0,2]$ on a real number line.
Now you want to find the probability of a number lying between $[0,1]$
Now,
Consider an event defined by the following equations and the occurrence of these events are governed by the above probability distribution in the range of $[0,2]$:
$A_n = [0, 2^{-\frac1n} )$. Therefore 
$$A_1 = [0,.5)$$
$$A_2 = [0,.71)$$
$$A_3 = [0,.8)$$
 and so on and as $n$ tends to infinity
$A_n = [0,1)$

Here $P(A_n)$ would be some function giving a value. This value is the probability of the occurrence of An. This would also give the probability of the number lying in the interval $[0,1]$.

This is my understanding of this theorem as I am also reading the same book.