# Squeeze Theorem/Sequences/Complex Numbers

## Theorem

Let $\sequence {a_n}$ be a null sequence in $\R$, that is:

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

Let $\sequence {z_n}$ be a sequence in $\C$.

Suppose $\sequence {a_n}$ dominates $\sequence {z_n}$.

That is:

- $\forall n \in \N: \cmod {z_n} \le a_n$

Then $\sequence {z_n}$ is a null sequence.

## Proof

\(\ds \forall n \in \N: \, \) | \(\ds \cmod {z_n}\) | \(\le\) | \(\ds a_n\) | Definition of Dominate (Analysis) | ||||||||||

\(\ds \forall n \in \N: \, \) | \(\ds a_n\) | \(\le\) | \(\ds \size {a_n}\) | Negative of Absolute Value | ||||||||||

\(\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \, \) | \(\, \ds n > N \implies \, \) | \(\ds \size {a_n}\) | \(<\) | \(\ds \epsilon\) | Definition of Null Sequence | |||||||||

\(\ds \leadsto \ \ \) | \(\ds \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: \, \) | \(\, \ds n > N \implies \, \) | \(\ds \cmod {z_n}\) | \(<\) | \(\ds \epsilon\) | Extended Transitivity |

Thus $\sequence {z_n}$ is a null sequence.

$\blacksquare$

## Also known as

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

In that culture, the word **sandwich** traditionally means specifically enclosing food between two slices of bread, as opposed to the looser usage of the **open sandwich**, where the there is only one such slice.

Hence, in idiomatic British English, one can refer to the (often uncomfortable) situation of being between two entities as being **sandwiched** between them.

As the idiom is not universal globally, the term **squeeze theorem** is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$, for greatest comprehension.