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?


Hint: $$\frac{1}{n}\ge\ln(n+1)-\ln(n)$$
