Math for Non-Geeks/The squeeze theorem

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

File:Beispielaufgabe zum Konvergenzbeweis einer Wurzelfolge mit dem Sandwichsatz.webm The intuition behind this theorem is quite simple: We are given a complicated sequence $(a_{n})_{n \in ℕ}$ and want to know whether it converges. Often, one can leave out terms in the complicated sequence $(a_{n})_{n \in ℕ}$ and gets some simpler sequences $(b_{n})_{n \in ℕ}$ and $(c_{n})_{n \in ℕ}$ . If $(b_{n})_{n \in ℕ}$ is a lower bond and $(c_{n})_{n \in ℕ}$ an upper bound, then $(a_{n})_{n \in ℕ}$ 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 $(a_{n})_{n \in ℕ}$ has no other option than to converge towards $a$ , as well.

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

The squeeze theorem

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

Math for Non-Geeks: Template:Satz

Math for Non-Geeks: Template:Hinweis

Math for Non-Geeks: Template:Beispiel

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 $(a_{n})_{n \in ℕ}$ , it can also be used for $a = 0$ . Especially, for any convergent $(a_{n})_{n \in ℕ}$ , the sequence $(| a_{n} - a |)_{n \in ℕ}$ must converge to 0: