Example 8 (2004 Jiangsu Summer Camp) Let $S=\frac{1}{\sqrt{1}+\sqrt{2}}+$ $\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{2003}+\sqrt{2004}}$, find the integer part of $S$.


\begin{array}{l}\text { Sol: For } a>0, b>0, \\ \text { it is easy to prove: } \frac{4}{\sqrt{a}+\sqrt{b}} \leqslant \frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}, \\ \text { thus } \frac{1}{\sqrt{n}+\sqrt{n+1}}\frac{1}{2}(\sqrt{2004}-1)=\sqrt{501}-0.5, \\ \text { hence } \therefore \sqrt{501}-0.5<S<\sqrt{501}, \\ \therefore 22-0.5<S<23, \text { therefore the integer part of } S \\ \text { is } 22.\end{array}
