# suppose that $a_1 \geq \dotsb \geq a_k$ and that $\lim_{k\to\infty} a_k=0$

Suppose that $a_1\geq \dotsb \geq a_k\geq \dotsb \geq 0$ and that $\lim_{k\to\infty} a_k = 0$.
Let $$S_N = \sum_{k=1}^N(-1)^{k+1}a_k.$$ Show that $$\left|\sum_{k=1}^\infty(-1)^{k+1}a_k-S_N\right|<a_N.$$

Source.

I'm stuck in this question and don't know how start any help please

• The claim is false. Try $a_1=a_2=1$, $a_3=a_4=\ldots = 0$ and $N=1$. – Hagen von Eitzen Jan 18 '16 at 12:43
• Formatting tips here. – Em. Jan 18 '16 at 12:44
• It clearly can't be true if any $a_k=0$, because it is possible for $|z|<0$. @HagenvonEitzen – Thomas Andrews Jan 18 '16 at 12:54
• how ???, as you see in question it asked about prove the next statement – user155971 Jan 18 '16 at 13:16
• I don't see why downvoting is warranted here. It looks like OP has stumbled upon this (ill-posed) excercise. – Roland Jan 18 '16 at 14:00