How to check whether a given inequality is correct for a large span of integers? The inequality
$\sqrt{n+ 1}−\sqrt n < \frac{1}{\sqrt n}$
is false for all n such that $101 ≤ n ≤ 2000$. Is the statement true?

Hint: Note that 
$$\sqrt{n+1}-\sqrt{n}=\frac{\left(\sqrt{n+1}-\sqrt{n}\right)\left(\sqrt{n+1}+\sqrt{n}\right)}{\sqrt{n+1}+\sqrt{n}}=\frac{1}{\sqrt{n+1}+\sqrt{n}}.$$
