1. Let $S=\sqrt{1+\frac{1}{1^{2}}+\frac{1}{2^{2}}}+\sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+$ $\cdots+\sqrt{1+\frac{1}{1999^{2}}+\frac{1}{2000^{2}}}$. Find the greatest integer not exceeding $S$.
(2000, Taiyuan Junior High School Mathematics Competition)
Prompt: Following Example 2, the largest integer not exceeding $S$ is
$$
1999 .
$$