Squeeze Theorem/Sequences

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Theorem

There are two versions of this result:


Sequences of Real Numbers

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

Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:

$\displaystyle \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$

Suppose that:

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


Then:

$x_n \to l$ as $n \to \infty$

that is:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$


Thus, if $\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\sequence {x_n} $ is said to be sandwiched or squeezed between those two sequences and itself must therefore converge to that same limit.


Sequences of Complex Numbers

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$ which is null, that is:

$a_n \to 0$ as $n \to \infty$

Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.


Suppose $\left \langle {a_n} \right \rangle$ dominates $\left \langle {z_n} \right \rangle$.

That is, suppose that:

$\forall n \in \N: \left|{z_n}\right| \le a_n$


Then $\left \langle {z_n} \right \rangle$ is a null sequence.


Sequences in a Linearly Ordered Space

Let $\left({S, \le, \tau}\right)$ be a linearly ordered space.

Let $\left\langle{x_n}\right\rangle$, $\left\langle{y_n}\right\rangle$, and $\left\langle{z_n}\right\rangle$ be sequences in $S$.

Let $p \in S$.

Let $\left\langle{x_n}\right\rangle$ and $\left\langle{z_n}\right\rangle$ both converge to $p$

For each $n$, let $x_n \le y_n \le z_n$.


Then $\left\langle{y_n}\right\rangle$ converges to $p$.


Sequences in a Metric Space

Let $M = \left({S, d}\right)$ be a metric space or pseudometric space.

Let $\left\langle{x_n}\right\rangle$ be a sequence in $S$.

Let $p \in S$.

Let $\left\langle{r_n}\right\rangle$ be a sequence in $\R_{\ge 0}$.

Let $\left\langle{r_n}\right\rangle$ converge to $0$.

For each $n$, let $d \left({p, x_n}\right) \le r_n$.


Then $\left\langle{x_n}\right\rangle$ converges to $p$.


Also known as

This result is also known, in the UK in particular, as the sandwich theorem.