Reading Körner's Companion to analysis there's an exercise that asks whether $c_n \rightarrow c$ is implied if both $a_n \rightarrow c$ and $b_n \rightarrow c$ when $|b_n| \leq |c_n| \leq |a_n|$.
Initially I thought that the implication should be false, since we could have $a_n \leq b_n \leq c_n$ (e.g. when $a_n < 0$ and $0 \leq b_n \leq c_n$), i.e. it could be that the sequence $\{c_n\}$ is not "squeezed" by $\{a_n\}$ and $\{b_n\}$. But I struggle to find an example.
However, if we are to have that $a_n \leq b_n \leq c_n$ and $|b_n| \leq |c_n| \leq |a_n|$, then I think we need to have $|c_n - b_n| < |a_n - b_n|$, which will imply that $c_n \rightarrow c$.
Could you help me either
- Find example sequences such that $a_n \leq b_n \leq c_n$ and $|b_n| \leq |c_n| \leq |a_n|$, where $a_n \rightarrow c$ and $b_n \rightarrow c$ but $c_n \nrightarrow c$.
- Or comment whether my argument above, using $|c_n - b_n| < |a_n - b_n|$, is correct.
Take $\{a_n\}$ and $\{b_n\}$ to be constant sequence, say $1$ and $\{c_n\}$ alternates between $1$ and $-1$. This serves as a counter example to the said claim.