<s>[INST] 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}$
 [/INST] This is a standard / common problem.
Hint: 
$$\prod_{i=1}^{98} \frac {i}{i+1} = \frac {1}{99}$$
$$\sqrt{\frac {1}{99}} \times \frac {99}{100} = \frac {\sqrt{99}}{100} < \frac {1}{10}$$

Hint elaborated slightly:
Let $A = \prod_{i=1}^{49} \frac {2i-1}{2i}$ and $B= \prod_{i=1}^{49} \frac {2i}{2i+1}$. It is clear that $A<B$. 
The question wants us to show that $A \times \frac {99}{100} < \frac {1}{10}$.
</s>