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squeeze rule (Theorem)

Squeeze rule for sequences

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

$\displaystyle f(n)\le g(n)\le h(n)$
for all $ n$. If $ \lim_{n\to\infty}f(n)$ and $ \lim_{n\to\infty}h(n)$ exist and are equal, say to $ a$, then $ \lim_{n\to\infty}g(n)$ also exists and equals $ a$.

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

$\displaystyle \vert a-f(n)\vert<\epsilon$    for all $\displaystyle n\ge M$
$\displaystyle \vert a-h(n)\vert<\epsilon$    for all $\displaystyle n\ge N$
Write $ L=\max(M,N)$. For $ n\ge L$ we have
  • if $ g(n)\ge a$:
    $\displaystyle \vert g(n)-a\vert=g(n)-a\le h(n)-a<\epsilon $
  • else $ g(n)<a$ and:
    $\displaystyle \vert g(n)-a\vert=a-g(n)\le a-f(n)<\epsilon $
So, for all $ n\ge L$, we have $ \vert g(n)-a\vert<\epsilon $, which is the desired conclusion.

Squeeze rule for functions

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

$\displaystyle f(x)\le g(x)\le h(x)$
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 exists and equals $ a$.

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

$\displaystyle \vert a-f(x)\vert<\epsilon$    whenever $\displaystyle 0<\vert b-x\vert<\alpha$
$\displaystyle \vert a-h(x)\vert<\epsilon$    whenever $\displaystyle 0<\vert b-x\vert<\beta$
Write $ \delta=\min(\alpha,\beta)$. Now, for any $ x$ such that $ \vert b-x\vert<\delta$, we have
  • if $ g(x)\ge a$:
    $\displaystyle \vert g(x)-a\vert=g(x)-a\le h(x)-a<\epsilon $
  • else $ g(x)<a$ and:
    $\displaystyle \vert g(x)-a\vert=a-g(x)\le a-f(x)<\epsilon $
and we are done.



"squeeze rule" is owned by Daume. [ owner history (1) ]
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Other names:  squeeze theorem, squeeze test
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Cross-references: positive, neighbourhood, functions, conclusion, hypothesis, real numbers, sequences
There are 3 references to this entry.

This is version 1 of squeeze rule, born on 2003-07-20.
Object id is 4483, canonical name is SqueezeRule.
Accessed 6599 times total.

Classification:
AMS MSC: 26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line)
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