người mù

\(\frac{1}{10}<\frac{1}{10.11}\)

\(\frac{1}{11}<\frac{1}{11.12}\)

\(...\)

\(\frac{1}{19}<\frac{1}{19.20}\)

\(\Rightarrow\frac{1}{10}+\frac{1}{11}+...+\frac{1}{19}<\frac{1}{10.11}+\frac{1}{11.12}+...+\frac{1}{19.20}=\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}+...+\frac{1}{19}-\frac{1}{20}\)

\(\frac{1}{10}+\frac{1}{11}+...+\frac{1}{11}<\frac{1}{10}-\frac{1}{20}\)

\(\frac{1}{10}+\frac{1}{11}+...+\frac{1}{19}<\frac{1}{20}\)

Mà \(\frac{1}{20}<\frac{1}{2}\) nên \(\frac{1}{10}+\frac{1}{11}+...+\frac{1}{19}<\frac{1}{2}\)