Can we apply squeeze in that way? Claim:
if $a_n\leq b_n\leq c_n$ for all $n\in \mathbb{N}$ and  $\displaystyle\sum\limits_{n=0}^{\infty} a_n,\displaystyle\sum\limits_{n=0}^{\infty} c_n$ are convergent then$\displaystyle\sum\limits_{n=0}^{\infty} b_n$ is convergent.
I think it is a wrong statement but I could not find any counterexample.
If you find a counterexample or prove it,I would be thankful.

$$\sum_{i=k}^l a_i \le  \sum_{i=k}^l b_i \le \sum_{i=k}^l c_i$$
This implies that
$$|\sum_{i=k}^l b_i| \le\max \{ |\sum_{i=k}^l a_i|, |\sum_{i=k}^l c_i|\}$$
Now $\sum a_i$, $\sum c_i$ exist, hence  and right term becomes arbitrarily small  (Cauchy-criterion),  if $k,l$  are chosen large enough, hence the partial sums $B_k:=\sum_0^k b_i$ are Cauchy as well.
