Problem

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Tags: algebra



(a) Prove that $\sqrt{n+1}-\sqrt n<\frac1{2\sqrt n}<\sqrt n-\sqrt{n-1}$ for all $n\in\mathbb N$. (b) Prove that the integer part of the sum $1+\frac1{\sqrt2}+\frac1{\sqrt3}+\ldots+\frac1{\sqrt{m^2}}$, where $m\in\mathbb N$, is either $2m-2$ or $2m-1$.