Squeeze Theorem/Sequences/Linearly Ordered Space

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Theorem

Let $\struct {S, \le, \tau}$ be a linearly ordered space.

Let $\sequence {x_n}$, $\sequence {y_n}$, and $\sequence {z_n}$ be sequences in $S$.

Let $p \in S$.

Let $\sequence {x_n}$ and $\sequence {z_n}$ both converge to $p$.

Let $\forall n \in \N: x_n \le y_n \le z_n$.

Then $\sequence {y_n}$ converges to $p$.

Proof

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

Then $\sequence {x_n}$ eventually succeeds $m$.

Thus by Extended Transitivity, $\sequence {y_n}$ eventually succeeds $m$.

A similar argument using $\sequence {z_n}$ proves the dual statement.

Thus $\sequence {y_n}$ is eventually in each ray containing $p$, so it converges to $p$.

$\blacksquare$