Đặt A=\(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+.......+\frac{1}{\sqrt{100}}\)
\(=\frac{2}{2\sqrt{1}}+\frac{2}{2\sqrt{2}}+\frac{2}{2\sqrt{3}}+.......+\frac{2}{2\sqrt{100}}\)
\(< 2\left(\sqrt{1}-\sqrt{0}+\sqrt{2}-\sqrt{1}+.........+\sqrt{100}-\sqrt{99}\right)\)
\(=2.\sqrt{100}=20\)
\(\Rightarrow A< 20\left(đpcm\right)\)