Ta có $\left\{ \begin{align}& \frac{1}{{{2}^{2}}} < \frac{1}{1.2} \\ & .............. \\ & \frac{1}{{{100}^{2}}} < \frac{1}{99.100} \\ \end{align} \right.$
$\Rightarrow F < \frac{1}{1.2}+\frac{1}{2.3}+...\frac{1}{99.100}$
$\Rightarrow F < \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}$
$\Rightarrow F < 1-\frac{1}{100}=\frac{99}{100}$
$\Rightarrow F < \frac{99}{100}\text{ (1)}$
Lại có $\left\{ \begin{align}& \frac{1}{{{2}^{2}}} > \frac{1}{2.3} \\ & .............. \\ & \frac{1}{{{100}^{2}}} > \frac{1}{100.101} \\ \end{align} \right.$
$\Rightarrow F > \frac{1}{2.3}+\frac{1}{3.4}+...\frac{1}{100.101}$
$\Rightarrow F > \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{100}-\frac{1}{101}$
$\Rightarrow F > \frac{1}{2}-\frac{1}{101}=\frac{99}{202}$
$\Rightarrow F > \frac{99}{202}\text{ (2)}$
Từ (1) và (2) $\Rightarrow \frac{99}{202} < F < \frac{99}{100}$