NL-NWO 1987 Problem 2

For $x >0$ , prove that $$\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}$$ and for all $n \ge 2$ prove that $$1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2$$