squeeze rule

Squeeze rule for sequences

Let $f,g,h:\mathbb{N}\to ℝ$ be three sequences of real numbers such that

$$f(n)\le g(n)\le h(n)$$

for all $n$. If ${lim}_{n\to \mathrm{\infty }}f(n)$ and ${lim}_{n\to \mathrm{\infty }}h(n)$ existand are equal, say to $a$, then ${lim}_{n\to \mathrm{\infty }}g(n)$ also exists andequals $a$.

The proof is fairly straightforward. Let $ϵ$ be any real number $>0$.By hypothesis there exist $M,N\in \mathbb{N}$ such that

Write $L=\max (M,N)$. For $n\ge L$ we have

• if $g(n)\ge a$:

• else and:

So, for all $n\ge L$, we have , which is the desired conclusion.

Squeeze rule for functions

Let $f,g,h:S\to ℝ$ be three real-valued functions on a neighbourhood$S$ of a real number $b$, such that

$$f(x)\le g(x)\le h(x)$$

for all $x\in S-\{b\}$. If ${lim}_{x\to b}f(x)$ and ${lim}_{x\to b}h(x)$exist and are equal, say to $a$, then ${lim}_{x\to b}g(x)$ also existsand equals $a$.

Again let $ϵ$ be an arbitrary positive real number. Find positive reals$\alpha$ and $\beta$ such that

Write $\delta =\min (\alpha ,\beta )$. Now, for any $x$ such that, we have

• if $g(x)\ge a$:

• else and:

and we are done.