I've got this question I can't even think how to begin answering...

Let $\displaystyle a_1, a_2, ... $ be a monotonically non-increasing sequence of positive numbers, so that the series $\displaystyle \sum_{n=1}^{\infty} a_n$ converges. Let S be the set of all numbers obtainable as $\displaystyle \sum_{j=1}^{\infty} a_{n_j}$ for some series $\displaystyle n_1 < n_2 < n_3 < ...$.

Show that S is an intervalifffor all $\displaystyle n \in N$,

$\displaystyle a_n \le \sum_{j=n+1}^{\infty} a_j$ .

For which $\displaystyle r > 0$ does $\displaystyle \sum r^n$ satisfy the condition?

Thanks in advance...