Squeeze Theorem/Sequences/Linearly Ordered Space

Theorem
Let $(S,\le,\tau)$ be a Linearly Ordered Space.

Let $\langle x_n \rangle$, $\langle y_n \rangle$, and $\langle z_n \rangle$ be sequences in $S$.

Let $p \in S$.

Suppose that $\langle x_n \rangle$ and $\langle z_n \rangle$ both converge to $p$

Suppose that for each $n$, $x_n \le y_n \le z_n$.

Then $\langle y_n \rangle$ converges to $p$.

Proof
Let $m \in S$ and $m < p$.

Then $\langle x_n \rangle$ eventually succeeds $m$.

Thus by Extended Transitivity, $\langle y_n \rangle$ eventually succeeds $m$.

A similar argument using $\langle z_n \rangle$ proves the dual statement.

Thus $\langle y_n \rangle$ is eventually in each ray containing $p$, so it converges to $p$.