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I would like to prove that given three sequences ${a_n}, {b_n}\text{ and }{c_n}$ and knowing that:

  1. They aren't necessarily of positive terms.
  2. $a_n \leq b_n \leq c_n, \forall n \geq 1$

$$\text{If }\sum_{n = 1}^{+ \infty}{a_n}\text{ and }\sum_{n = 1}^{+ \infty}{c_n}\text{ are both convergent then }\sum_{n = 1}^{+ \infty}{b_n}\text{ converges and}$$ $$\sum_{n = 1}^{+ \infty}{a_n} \leq \sum_{n = 1}^{+ \infty}{b_n} \leq \sum_{n = 1}^{+ \infty}{c_n}$$

However I'm having difficulties. This is what I've done so far:

$$\text{We have that } \sum_{n = 1}^{+ \infty}{a_n}\text{ converges so we know that } \lim_{N\to{+ \infty}}{A_N} = L$$ $A_N$ is the sequence of partial sums.

$$\text{The same holds for }\sum_{n = 1}^{+ \infty}{c_n}\text{;} \lim_{N\to{+ \infty}}{C_N} = L'$$

$$\text{So }L \leq \lim_{N\to{+ \infty}}{B_N} \geq L'$$

Because the limit of $B_N$ is between $L$ and $L'$ then I can say that $\sum_{n = 1}^{+ \infty}{b_n}$ converges.

The thing is I'm not sure about the assertion, moreover the proof looks easy this way, which makes me suspect.

$$\text{The other thing is: because }\sum_{n = 1}^{+ \infty}{a_n}\text{ and }\sum_{n = 1}^{+ \infty}{c_n}\text{ are both convergent, then }\lim_{n\to{+ \infty}}{a_n} = \lim_{n\to{+ \infty}}{c_n} = 0\text{, so by the sandwich principle }\lim_{n\to{+ \infty}}{b_n} = 0\text{, but that of course doesn't allow me to assert that }\sum_{n = 1}^{+ \infty}{b_n}\text{ converges.}$$

Hope you could help me. Thanks.

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2 Answers 2

up vote 1 down vote accepted

We have $0\le b_n-a_n\le c_n-a_n$. The series $\sum_{n=1}^\infty(c_n-a_n)$ is convergent because both $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty c_n$ are; moreover, it's terms are positive. The comparison test for series of positive terms proves that $\sum_{n=1}^\infty(b_n-a_n)$ is convergent. Since $b_n=a_n+(b_n-a_n)$, it follows that $\sum_{n=1}^\infty b_n$ is convergent.

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Nice solution.. –  N. S. Jun 13 '12 at 16:33
    
I was left speechless by this elegant solution. Thank you. –  David Robert Jones Jun 13 '12 at 18:56

Hint: Let $n <m$. Then

$$ \sum_{k=n}^m a_n \leq \sum_{k=n}^m b_n \leq \sum_{k=n}^m c_n \,.$$

Using the convergence of $\sum_{k=0}^\infty a_n$ and $\sum_{k=0}^\infty c_n$, you can prove that $$\sum_{k=0}^\infty b_n$ is Cauchy.

Just be carefull with the fact that things could be positive or negative, it is very easy to get around this problem....

Let me know if you run into trouble so I can give more details ;)

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