> [!theorem] > > Let $x_n, y_n, z_n$ be three [[Sequence|sequences]] where > $ > x_n \le y_n \le z_n \forall n \in \nat^{+} > $ > and $\limv{n}x_n = \limv{n}z_n = e$, then, the [[Limit|limit]] $\limv{n}y_n = e$. > > *Proof*. Let $\varepsilon \gt 0$, $N_1 \in \nat^+: |z_n - e| \lt \varepsilon \forall n \ge N_1$, and $N_2 \in \nat^+: |x_n - e| \lt \varepsilon \forall n \ge N_2$. Take $N = \max(N_1, N_2)$. Then > $ > \begin{align*} > y_n - e &\le z_n - e \lt \varepsilon &\forall n \ge N \\ > y_n - e &\ge x_n - e \gt -\varepsilon &\forall n \ge N > \end{align*} > $ > Therefore, $|y_n - e| \lt \varepsilon$. Hence, $\limv{n}y_n = e$.