Math Help - Convergence

1. Convergence

Let $\{a_n\}$ be a strictly increasing sequence of positive integers.
Can the series
$\sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots$
ever converge?

2. Originally Posted by math sucks
Let $\{a_n\}$ be a strictly increasing sequence of positive integers.
Can the series
$\sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots$
ever converge?
At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
$\sum_{n = 1}^{\infty}\frac{1}{n}$
which is divergent.

-Dan

3. Originally Posted by topsquark
At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
$\sum_{n = 1}^{\infty}\frac{1}{n}$
which is divergent.

-Dan
Can you prove your statement?