Just in case your were wondering, you don't need LaTeX for absolute value, the key is right above your Enter key on your keyboard (PC): 
So we know [itex]\left\frac{a_{n+1}}{a_n}\right < r[/itex] which is the same as [itex]a_{n+1} < r a_n[/itex]. This is saying that the n+1st term (the term after the nth term) in the sequence is the nth term multiplied by r.
What they are doing in the proof is going backwards. Think of it this way, it might help you better visualize it:
Starting with [itex]a_{N+1} < r a_N[/itex] and [itex]a_{N+2} < r a_{N+1}[/itex] we can substitute for [itex]a_{N+1}[/itex] below:
[itex]
\begin{eqnarray*}
a_{N+2} &<& r a_{N+1}\\
&<& r (r a_{N}) = r^2 a_N
\end{eqnarray*}
[/itex]
And you can continue this process
[itex]
\begin{eqnarray*}
a_{N+3} &<& r a_{N+2}\\
&<& r (r^2 a_{N}) = r^3 a_N
\end{eqnarray*}
[/itex]
Does that help or is it still confusing?
