# Prove that: $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\ge\frac{2}{3}$

I've got three inequalities: $\forall n\in\mathbb N:$

$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{1}{2}$$

$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{7}{12}$$

$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\ge\frac{2}{3}$$

From what I know the LHS converges to something about $0.69$ and each one of them requires the same method, but I can't come up with a proper way to solve it.

Can someone give me a hint?

• You know that you only need to prove one inequality here, and not "three"? – Dietrich Burde Oct 7 '17 at 11:31