Let 
$$a\_n=\frac1{\sqrt1}+\frac1{\sqrt 2}+\ldots +\frac1{\sqrt n}-2\sqrt n $$
For the task to prove that $$\tag1-2\le a\_n\le -1 $$
I was given the hint
$$\tag2\sqrt{k+1}-\sqrt k<\frac1{2\sqrt k},\qquad \forall k\in\mathbb N^\* $$
I managed to prove $(1)$ by other methods but woonder how to proof it actually using $(2)$.