Cómo probar $\sqrt{1000} < x < 1000$ ?

Question

Me han dado que

$$x = \frac 21 \times \frac 43 \times \frac 65 \times \frac 87 \times \cdots \times \frac {996}{995} \times \frac{998}{997} \times \frac {1000}{999}$$

¿Cómo puedo demostrar que $\sqrt{1000} < x < 1000$ ?

Answer 1

\begin {align} x^2 &= \left ( \frac 21 \times \frac 21 \right ) \times \left ( \frac 43 \times \frac 43 \right ) \times \cdots \times \left ( \frac {1000}{999} \times \frac {1000}{999} \right ) \\ & \ge \left ( \frac 21 \times \frac 32 \right ) \times \left ( \frac 43 \times \frac 54 \right ) \times \cdots \times \left ( \frac {1000}{999} \times \frac {1001}{1000} \right ) \\ &= 1001 \end {align}