Prove that a series is bounded with induction I have to prove that the following condition is true:
$$\frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2n} > \frac{13}{24}$$
for every $n > 1$.
I understood that this series is the same as:
$$S(n) = \sum_{i=1}^n \frac{1}{n + i} $$
I tried to use induction but I could't reach something meaningful. 
(This exercise was in a discrete math exam, so I am not looking for a proof using series theory).
Thanks!

Let
$$
S(n) = \frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n}.
$$
So,
$$
S(n+1) = \frac{1}{n+2} + \frac{1}{n+3} + \ldots + \frac{1}{2n} + \frac{1}{2n+1} + \frac{1}{2n+2}
$$
and
$$
S(n+1) = S(n) - \frac{1}{n+1} + \frac{1}{2n+1} + \frac{1}{2n+2} = S(n) + \frac{1}{2n+1} - \frac{1}{2n+2} =\\= S(n) + \frac{1}{(2n+1)(2n+2)}
$$
For $n=2$ we have
$$
S(2) = \frac13 + \frac14 = \frac{7}{12} = \frac{14}{24} > \frac{13}{24},
$$
and $S(n)$ strictly increasing. So...?
