Question 167
We consider the numbers
$$
A = \frac{1}{4} \cdot \frac{3}{6} \cdot \frac{5}{8} \cdots \frac{595}{598} \cdot \frac{597}{600} \text{ and } B = \frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{9} \cdots \frac{596}{599} \cdot \frac{598}{601}.
$$
Prove that: (a) $A \lt B$, (b) $A \lt \frac{1}{5990}$.
$$
A = \frac{1}{4} \cdot \frac{3}{6} \cdot \frac{5}{8} \cdots \frac{595}{598} \cdot \frac{597}{600} \text{ and } B = \frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{9} \cdots \frac{596}{599} \cdot \frac{598}{601}.
$$
Prove that: (a) $A \lt B$, (b) $A \lt \frac{1}{5990}$.