Solve this inequality $\prod_{i=1}^{50} \frac {2i-1}{2i} < \frac {1}{10}$ Prove that
$ \frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \frac{7}{8}\cdot \frac{9}{10}\cdot \frac{11}{12}\cdot \frac{13}{14}...\cdot \frac{91}{92}\cdot \frac{93}{94}\cdot \frac{95}{96}\frac{97}{98}\cdot \frac{99}{100} <\frac{1}{10}$

The product of the first 31 fractions is $0.100923...$ while the product of the first 32 fractions is $0.099346...$, so it looks like you only need the first 32 of the 50 fractions to get below $1/10$. Using more fractions (all less than 1) will only make that smaller.
