I would like to prove that given three sequences ${a_n}, {b_n}\text{ and }{c_n}$ and knowing that:
- They aren't necessarily of positive terms.
- $a_n \leq b_n \leq c_n, \forall n \geq 1$
$\text{If }\sum_{n = 1}^{+ \infty}{a_n}\text{ and }\sum_{n = 1}^{+ \infty}{c_n}\text{ are both convergent then }\sum_{n = 1}^{+ \infty}{b_n}\text{ converges and}$ $\sum_{n = 1}^{+ \infty}{a_n} \leq \sum_{n = 1}^{+ \infty}{b_n} \leq \sum_{n = 1}^{+ \infty}{c_n}$
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.