\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)
\(=\left(\frac{1}{51}+\frac{1}{52}+....+\frac{1}{75}\right)+...+\left(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}\right)\)
\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}< \frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\)
\(\frac{1}{51}+\frac{1}{52}+..+\frac{1}{75}< \frac{1}{2}\)
\(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}< \frac{1}{75}+\frac{1}{75}+\frac{1}{75}+...+\frac{1}{75}=\frac{1}{3}\)
\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}< \frac{1}{2}+\frac{1}{3}=\frac{5}{6}< 1\)
\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}>\frac{1}{75}+\frac{1}{75}+...+\frac{1}{75}\)
\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}>\frac{1}{3}\)
\(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}>\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}\)
\(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}>\frac{1}{4}\)
\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}>\frac{1}{3}+\frac{1}{4}=\frac{7}{12}>\frac{6}{12}=\frac{1}{2}\)
