How to prove $\sqrt{1000} < x < 1000$?

Question

I have been given that

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

How can I prove that $\sqrt{1000} < x < 1000$?

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}