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Let $(a_n),(b_n),(c_n)$ be sequences of real numbers with the property that for each $n\in\mathbb{N}$, $a_n\leq b_n\leq c_n$. Suppose both series, $\sum\limits_{n=1}^\infty a_n$ and $\sum\limits_{n=0}^\infty c_n$ converge. Then prove that $\sum\limits_{n=0}^\infty b_n$ also converge.

I read the similar post in here. But couldn't relate it to my question. Appreciate any help.

DD90
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  • Sorry. I edited the question – DD90 Jan 07 '19 at 23:13
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    Have you tried to use the Cauchy sequence criterion? – Aphelli Jan 07 '19 at 23:14
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    @Mindlack gives a good hint. If the sum of $a_n$'s and $b_n's$ converged to the same thing, there would be an obvious candidate for the sum of the $b_n's$. However, you're not given that, so the sum of the $b_n's$ can kind of do whatever it wants as long as it stays between the bounds. So it is in this case, that you have no idea what the sum would converge to, that the Cauchy cirterion is taylored for. – Ovi Jan 07 '19 at 23:17
  • Note that completeness is needed; in $\mathbb{Q}$, $\dfrac{1}{n^2+n}\le\dfrac{1}{n^2}\le\dfrac{1}{n^2-n}$ gives a counterexample. – Jianing Song Oct 09 '23 at 21:37

1 Answers1

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This is an immediate corollary to the comparison test. Consider $$0 \le b_n - a_n \le c_n - a_n.$$ Then, $\sum (c_n - a_n)$ is a convergent, positive series, hence so is $\sum (b_n - a_n)$ (the partial sums of $\sum (b_n - a_n)$ form a monotone sequence, bounded above by $\sum (c_n - a_n)$; apply monotone convergence theorem). Since $\sum a_n$ is convergent, then so is $\sum b_n = \sum (b_n - a_n) + \sum a_n$.

Theo Bendit
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  • So here does $b_n$ means the partial sums of the $\sum b_n$? – DD90 Jan 07 '19 at 23:19
  • No, just the individual terms. The individual terms of the series $\sum b_n - a_n$ are positive and dominated by the individual terms of the convergent series $\sum c_n - a_n$. – Theo Bendit Jan 07 '19 at 23:21
  • Ah! got it. It's a nice argument. Just to make it clear, comparison test is only valid for non negative terms.. is it? – DD90 Jan 07 '19 at 23:24
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    Yes, that's right. It's proven (as I alluded to in the answer) with the monotone convergence theorem, which requires positive terms in order to make the sequence of partial sums monotone. Essentially, your question is the non-positive version; it's ok to have negative terms, provided the terms are bounded below by a convergent (possibly negative) series. – Theo Bendit Jan 07 '19 at 23:28
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    Thank you! Appreciate your help – DD90 Jan 07 '19 at 23:30
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    "$\sum c_n - a_n$" Better to write $\sum (c_n - a_n)$ – zhw. Jan 07 '19 at 23:34