<s>[INST] Show $|a\sqrt{b}-\sqrt{c}|$ equal to zero or larger than $\frac{1}{2}10^{-3}$ when $a, b$ and $c$ are natural numbers strictly less than 100 I need to show that $|a\sqrt{b}-\sqrt{c}|$ is equal to zero or larger than $\frac{1}{2}10^{-3}$ when $a, b$ and $c$ are natural numbers strictly less than 100. 
I see why it can be equal to zero. It is the other part that is causing my trouble. I tried to consider $(a\sqrt{b}-\sqrt{c})(a\sqrt{b}+\sqrt{c})=a^2b-c$, but I really don't know if it helps. Any hints would be appreciated.
Thanks!
 [/INST] If $a\sqrt{b} - \sqrt{c} \neq 0$, then $|a^2b - c| \geq 1$. We see that:
$$
a\sqrt{b} + \sqrt{c} < 100(10) + 10 = 1010
$$
Therefore:
$$
|a\sqrt{b} - \sqrt{c}| = \frac{|a^2b - c|}{a\sqrt{b} + \sqrt{c}} > \frac{1}{1010} > \frac{1}{2000}
$$
</s>