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The inequality $\sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n }$ is false for all in n such that $101 \le n \le 2000$

False

Discussion:

$\sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n }$

By cross multiplying we have  $\sqrt {(n+1)n} - n < 1$. That is $\sqrt {n(n+1)} < (n+1)$  or $n(n+1) < (n+1) ^2$ or  n < n+1

This is true for all n.