Squeeze Theorem/Sequences/Complex Numbers

From ProofWiki
Jump to navigation Jump to search

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.

Retrieved from "https://proofwiki.org/w/index.php?title=Squeeze_Theorem/Sequences/Complex_Numbers&oldid=528765"