TITLE: Sum of inverse of two divergent sequences
QUESTION [7 upvotes]: Let $a_n, b_n$ be two real sequences for wich $0<a_n<b_n<a_{n+1}$. Let $\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=\infty$
I need to prove that $\sum{\frac{1}{a_n}-\frac{1}{b_n}}$ is convergent.
It's been a long time since I did some real analysis, so I probably forgot some basic thing to prove this, because I don't think that'll be very hard.
PS: English isn't my main language, forgive me some errors.

REPLY [7 votes]: Let $c_n$ be the sequence $c_1=\frac{1}{a_1}$, $c_2=\frac{1}{b_1}$,$c_3=\frac{1}{a_2}$, $c_4=\frac{1}{b_2}$, etc.
Its terms are all positive, the sequence is monotone decreasing, and the sequence converges to $0$. Applying the Leibniz criterion, we reach the slightly stronger conclusion that the series
$$S=\sum(-1)^{n+1}c_n$$
converges. This implies the convergence of your desired series in particular (the partial sums of your series are a subsequence of the partial sums of $S$).