Prove:
The sequence An = 1/(n + 1) + 1/(n + 2) +...+ 1/(2n) has a limit.
$\displaystyle n + 1 < n + 2 < \cdots < 2n \Leftrightarrow \frac{1}{2n} < \frac{1}{2n - 1} < \cdots < \frac{1}{n+1} $.
Notice that your sum has n amount of terms, and so $\displaystyle 0 < \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} < \frac{n}{2n} = \frac{1}{2} $.
Now show that it's monotonic, and you're finished.