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Given a sequence $a_n$ such $$ a_2 \leq a_4 \leq a_6 \le \cdots\cdots \leq a_5 \leq a_3 \leq a_1 $$ and a sequence ${b_n}$, where $b_n = a_{2n-1} - a_{2n}$ converges to $0$, then show that $a_n$ is convergent.

I can see that sequence $a_n$ has two subsequences (odd terms and even terms) which are decreasing. But any hints to get me furthure? Thanks.

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    $\begingroup$ The odd sequence is decreasing and bounded from below, thus converges to some $a$. The condition on $b_n$ just shows that the even term would also be closed to $a$ as $n$ large. $\endgroup$ – user99914 Nov 17 '14 at 5:57
  • $\begingroup$ @John Odd sequence is decreasing but how can u say that it is bounded from below? $\endgroup$ – godonichia Nov 17 '14 at 6:00
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    $\begingroup$ From what you wrote I thought it is bounded by $a_2$? $\endgroup$ – user99914 Nov 17 '14 at 6:18
  • $\begingroup$ @John how can condition on b says about even terms ? $\endgroup$ – godonichia Nov 17 '14 at 7:18
  • $\begingroup$ Are you familiar with Cauchy sequences and completeness, i.e. do you know that in $\mathbb{R}$ every Cauchy sequence converges? $\endgroup$ – J.A.L Nov 17 '14 at 7:21

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