Let $\{x_n\}$ be monotone increasing sequence of positive real numbers. Show that if $\{x_n\}$ is bounded, then $\sum_{n=1}^{\infty}\left(1-\frac{x_n}{x_{n+1}}\right)$ converges. On the other hand, if the sequence is unbounded, the series is divergent.
The following is the proof given by my lecturer, but I do not understand the proof:
Let $u_m = x_{m+1} -x_m$ and $d_m = \sum_{k=1}^m u_k = x_{m+1} - x_1$ and hence $x_{m+1} = d_m + x_1$
$$\sum_{n=1}^{\infty}\left(1-\frac{x_n}{x_{n+1}}\right) = \sum_{n=1}^{\infty}\frac{u_m}{d_m +x_1} $$
Then he says that if $d_m$ diverges, then the sum diverges. If $d_m$ converges, then then sum converges.
He says that it has something to do with Gauss Test.
Can someone explain the proof to me. Or does anyone has a better proof.