Theorem: Squeezing Theorem for Functions

Let $D\subseteq\mathbb R$ be a subset of real numbers and let $f,g,h:D\to\mathbb R$ be functions with $g(x)\le f(x)\le h(x)$ for all $x\in D$. Let $g,h$ have a limit $L$ at $x=a\in D.$ Then $f$ has also the limit $L$ at $x=a\in D.$ Formally, $$\left\lbrace\begin{array}{l}\lim_{x\to a}g(x)=L\\\lim_{x\to a} h(x)=L\\g(x)\le f(x)\le h(x)\;\forall x\in D\end{array} \right\rbrace\Longrightarrow \lim_{x\to a} f(x)=L.$$ This is what is known as the Squeezing Theorem of the Sandwich Theorem or the Scrunch Theorem.

Proofs: 1

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  1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016