Assume that for all \(n \in \mathbb{N}\) we have that \(x_{n}\leq y_{n}\leq z_{n}\) and that \(\lim_{ n \to \infty }x_{n}=\lim_{ n \to \infty }z_{n}=l\). Then we have that \(\lim_{ n \to \infty }y_{n=l}\).
Proof: From our assumptions we have that for all \(\epsilon >0\) \[ \begin{align} \forall n\geq N:=N_{x} \vee N_{y},\begin{cases} |x_{n}-l|<\epsilon \\ |z_{n}-l|<\epsilon \end{cases} \implies \begin{cases} x_{n}-l<\epsilon \\ l-x_{n}<\epsilon \\ z_{n}-l<\epsilon \\ l-z_{n}<\epsilon. \end{cases} . \end{align} \] From this and our assumption that \(x_{n}\leq y_{n}\leq z_{n}\) we have that \[ \begin{cases} y_{n}- l\leq z_{n}-l<\epsilon \\ l-y_{n}\leq l-x_{n}<\epsilon \end{cases}\implies\mid y_{n}-l|<\epsilon . \]