# Prove that for all $N\ge1$ and all $n\ge1$, $\sum_{k=1}^{n}\frac{a_{N+k}}{s_{N+k}} \ge 1 - \frac{s_N}{s_{N+n}}$

Let $$a_n$$ be a sequence of non-negative real numbers, such that $$\sum a_n$$ diverges. For $$n\ge1$$, let $$s_n = a_1 + ... + a_n$$.

Prove that for all $$N\ge1$$ and all $$n\ge1$$, $$\sum_{k=1}^{n}\frac{a_{N+k}}{s_{N+k}} \ge 1 - \frac{s_N}{s_{N+n}}$$

I tried to manipulate the terms to apply comparison test, but I couldn't really get anywhere with this problem. Thanks

• Have you tried induction? On the surface, this looks like an induction problem. – Clayton Nov 8 '18 at 19:22
• It seems I couldn't fit this question in full to the title due to character limits. – davidh Nov 8 '18 at 19:25
• oh, I misunderstood... will edit. – davidh Nov 8 '18 at 19:26