Prove that $1/n+1/(n+1)+\dots+(1/2n)>2/3$ 
Prove that $\displaystyle\frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n}>\frac{2}{3}$

I tried to use mathematical induction, but I'm not able to prove that:
$$
\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} + \frac{1}{2n+1} + \frac{1}{2n+2}>\frac{2}{3}.
$$
My method was:
Assumption:
$$\frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n}>\frac{2}{3}$$
$$\frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} + \frac{1}{2n+1} + \frac{1}{2n+2}>\frac{2}{3} + \frac{1}{2n+1} + \frac{1}{2n+2}$$
$$\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} + \frac{1}{2n+1} + \frac{1}{2n+2}>\frac{2}{3} + \frac{1}{2n+1} + \frac{1}{2n+2}-\frac{1}{n}$$
Now in order to prove the thesis I have to prove that
$$\frac{2}{3} + \frac{1}{2n+1} + \frac{1}{2n+2}-\frac{1}{n} > \frac{2}{3}$$
But it's a contradiction. Did I make a mistake somewhere? How can I solve this problem? I'd appreciate your help.

HINT
As suggested by Wojowu in the comment, sometimes induction works for a stronger hypothesis, in that case let try with
$$\displaystyle\frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n}>\frac{2}{3}+\frac1{4n}>\frac23$$
Refer also to the related


*

*Proof by induction, $1/2 + ... + n/2^n < 2$

*If $x>0$ real number and $n>1$ integer, then $(1+x)^n>\frac{1}{2}n(n-1)x^2$

*Show that $\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)...\left(1+\frac{1}{n^3}\right) < 3$
