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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

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

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