Prove that:
1 + 1/2 + 1/3 + ... + 1/n > 2n/n+1
where n > 2
This will be a proof by induction
For the base case let n=2. $\displaystyle 1+\frac{1}{2}>\frac{2(2)}{2+1}$
So $\displaystyle \frac{3}{2}>\frac{4}{3}$ which is true so the base case holds
Now we'll assume the statement is true for n and consider it for n+1
$\displaystyle \frac{1}{n+1}+\sum_{k=1}^n \frac{1}{k}>\frac{1}{n+1}+\frac{2n}{n+1}=\frac{2n+ 1}{n+1}$
$\displaystyle \frac{2(n+1)}{(n+1)+1}=\frac{2n+2}{n+2}$
Is $\displaystyle \frac{2n+1}{n+1}>\frac{2n+2}{n+2}$
$\displaystyle (2n+1)(n+2)=(2n^2+5n+2)$
$\displaystyle (2n+2)(n+1)=(2n^2+4n+2)$ (this is an analysis via cross multiplication)
And since 5n>4n for all n, $\displaystyle \frac{2n+1}{n+1}>\frac{2n+2}{n+2}$ so the statement is proven by induction