Example 5 Let $a=\frac{20052005}{20062006}, b=\frac{20062006}{20072007}$, $c=\frac{20072007}{20082008}$.
Try to compare the sizes of $a$, $b$, and $c$.
(2007, Nanning Junior High School Mathematics Competition)
Solve by dividing 1 by $a$ we get
$$
\begin{array}{l}
\frac{1}{a}=\frac{20062006}{20052005} \\
=\frac{2006 \times 10001}{2005 \times 10001} \\
=\frac{2006}{2005}=1+\frac{1}{2005} .
\end{array}
$$

Similarly, $\frac{1}{b}=1+\frac{1}{2006}, \frac{1}{c}=1+\frac{1}{2007}$.
Therefore, $\frac{1}{a}>\frac{1}{b}>\frac{1}{c} \Rightarrow a<b<c$.