# Need a hint to start prove an inequality [duplicate]

wan't know how to start this problem, tried to multiply by $\sqrt{n}$, but it does not work. Can someone help me with that? Thanks. Here is the inequality. $$2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$$

• – Martin R Jun 22 '17 at 5:01
• hint: use induction – MAN-MADE Jun 22 '17 at 6:25
• hint: use integration on $1/\sqrt{x}$ – MAN-MADE Jun 22 '17 at 6:27

Hint $$\sqrt{n+1}-\sqrt{n}=(\sqrt{n+1}-\sqrt{n})\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}\\ \sqrt{n}-\sqrt{n-1}=(\sqrt{n}-\sqrt{n-1})\frac{\sqrt{n}+\sqrt{n-1}}{\sqrt{n}+\sqrt{n-1}}$$
$$\sqrt {n}-\sqrt {n-1}=\frac {1}{2\sqrt {c}}$$
$$\sqrt {n+1}-\sqrt {n}=\frac {1}{2\sqrt {d}}$$ with $$n-1 <c <n<d <n+1 .$$