"Squeeze theorem" for series

Question

I am considering the following statement:

Let $a_n$, $b_n$ and $c_n$ be three sequences such that $a_n\leqslant b_n\leqslant c_n$ for any $n$. If $\sum a_n$ and $\sum c_n$ are convergent, then $\sum b_n$ is also convergent.

Question : Is the statement true?

My thoughts on the question:

1. If $b_n\geqslant 0$, then the statement is obviously true by Comparison Test.
2. Clearly, $b_n$ has $0$ for limit, by the Squeeze Theorem for sequences.
3. If $\sum a_n$ and $\sum c_n$ are absolutely convergent, then $\sum b_n$ is absolutely convergent as well, since $|b_n|\leqslant Max\{|a_n|,|c_n|\}\leqslant |a_n|+|c_n|$.

I fail to see a proof of the statement, but I cannot find a counter-example. Any hint?

Answer 1

Hint: Try subtracting $a_n$ from each of your sequences, to reduce to the case that $b_n\geq 0$ for all $n$.