Serlo: EN: The squeeze theorem

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The squeeze theorem is a powerful tool to determine the limit of a complicated sequence. It is based on comparison to simpler sequences, for which the limit is easily determinable.

Motivation

Datei:Beispielaufgabe zum Konvergenzbeweis einer Wurzelfolge mit dem Sandwichsatz.webm The intuition behind this theorem is quite simple: We are given a complicated sequence $(a_n{)}_{n\in \mathbb{N}}$ and want to know whether it converges. Often, one can leave out terms in the complicated sequence $(a_n{)}_{n\in \mathbb{N}}$ and gets some simpler sequences $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$. If $(b_n{)}_{n\in \mathbb{N}}$ is a lower bond and $(c_n{)}_{n\in \mathbb{N}}$ an upper bound, then $(a_n{)}_{n\in \mathbb{N}}$ is "caught" in the space between both functions. If both sequences converge to the same limit $a$, they "squeeze" together this space to this single point and $(a_n{)}_{n\in \mathbb{N}}$ has no other option than to converge towards $a$, as well.

image about the principle of the sandwich lemma

You may also visualize this theorem by a "ham & cheese sandwich" (see image on the right). The upper and lower bounds $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ act as two "slices of toast", which confine $(a_n{)}_{n\in \mathbb{N}}$ , i.e. the "filling". If you squeeze the toast slices together, the filling in between will also squeezed to this point.

The squeeze theorem

Datei:Sandwich Theorem - Beweis Anwendung Beispielaufgabe.webm The theorem reads as follows:

A useful special case

We often encounter 0 as a sequence limit (null sequence). Since the squeeze theorem can be used to prove any $a$ to be a limit of a sequence $(a_n{)}_{n\in \mathbb{N}}$, it can also be used for $a=0$. Especially, for any convergent $(a_n{)}_{n\in \mathbb{N}}$, the sequence $(|a_n-a|{)}_{n\in \mathbb{N}}$ must converge to 0:

Examples & exercises: squeeze theorem for null sequences

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