Let the real sequence ${x_n}$ be given by,
$$\sum_{j=1}^{2n} \frac {1}{j} - \sum_{j=1}^{n} \frac {1}{j}. $$
Show that $0<x_{n}<x_{n+1}$ and that $x_{n}<1$ for all $n$. Deduce that $x_{n}$ converges, giving your reason.
I seem to think this has something to do with $\sum_{j=1}^{2n} \frac {1}{j} - \sum_{j=1}^{n} \frac {1}{j} $ = $\sum_{j=1}^{n} \frac {1}{j+n}$?
Many thanks in advance.
First note that the sequence $(x_n)$ is bounded above. This follows from your observation that $\sum_{j=1}^{n} \frac {1}{j+n}$. Here we have $n$ terms, all of them clearly less than $1/n$, so their sum is less than $1$.
Next you want to show that the sequence $(x_n)$ is increasing. Calculate $x_{n+1}-x_n$, and show it is positive. Most of the terms cancel: $$x_{n+1}-x_n=\frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1}.$$ Finally, appeal to the theorem that an increasing sequence which is bounded above has a limit.
Remark: The limit is in fact $\ln(2)$, but it seems you are not asked to show that. If you wish, it can be done by a Riemann sum argument.
