In
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
, the
squeeze theorem (known also as the
pinching theorem, the
sandwich theorem, the
sandwich rule and sometimes the
squeeze lemma) is a
theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
regarding the
limit of a function.
The squeeze theorem is a technical result which is very important in proofs in
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
and
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.
In
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
, the
squeeze theorem (known also as the
pinching theorem, the
sandwich theorem, the
sandwich rule and sometimes the
squeeze lemma) is a
theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
regarding the
limit of a function.
The squeeze theorem is a technical result which is very important in proofs in
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
and
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
s
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity...
and
EudoxusEudoxus was the name of two ancient Greeks:* Eudoxus of Cnidus , Greek astronomer and mathematician.* Eudoxus of Cyzicus , Greek navigator....
in an effort to compute
πPi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706...
, and was formulated in modern terms by
GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics...
.
In
ItalyItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
,
RussiaRussia , officially known as both Russia and the Russian Federation , is a country in northern Eurasia . It is a semi-presidential republic, comprising 83 federal subjects...
and
FranceFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
, the squeeze theorem is also known as the
two carabinieriThe Arma dei Carabinieri is the national gendarmerie of Italy, policing both the military and civilian populations. The Carabinieri is now a branch of the armed forces , thus ending their long standing role as the first corps of the Italian army.-Early history:The corps was created by King Victor...
theorem,
two militsioner theorem or
two gendarmesA gendarmerie or gendarmery is a military body charged with police duties among civilian populations. The members of such a body are called gendarmes. The term maréchaussée may also be used but is now uncommon.-Etymology:The word gendarme comes from Old French gens d'armes, meaning men-at-arms...
theorem. The story is that if two police officers are holding a prisoner between them, and both the officers are going to the cells, the prisoner must also be going to the cells.
Statement
The squeeze theorem is formally stated as follows.
Let I be an intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
containing the point a. Let f, g, and h be functionsIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:
and also suppose that:
Then
- The functions g(x) and h(x) are said to be lower and upper bounds
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
(respectively) of f(x).
- Here a is not required to lie in the interior
In mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.
- A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.
Proof. From the above hypotheses we have, taking the limit inferior and superior:
so all the inequalities are indeed equalities and the thesis immediately follows.
Proof
The main idea behind this proof is to consider the
relative differences between the functions
f,
g, and
h. This has the effect of making the lower bound identically 0, and all the functions non-negative. This greatly simplifies the details of the proof. The general case then follows algebraically.
To begin the proof, assume all the hypotheses and notation as given in the statement of the theorem above. We first prove the special case where
g(
x) = 0 for all
x and
L = 0. In this case:
Let ε > 0 be any fixed positive number. By the definition of the
limit of a functionIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Informally, a function assigns an output f to every input x. The function has a limit L at an input p if f is "close" to L whenever x is...
, there is a δ > 0 such that:
For any
x in
I not equal to
a:
so that:
We conclude that:
This proves that:
This completes the proof for the special case. Now, we prove the general theorem by letting
g and
L be arbitrary. For any
x in
I not equal to
a, we have:
Subtracting
g(
x) from each expression:
As and , so that:
The special case now shows that We conclude that:
Q.E.D.Q.E.D. is an abbreviation of the Latin phrase , which literally means "which was to be demonstrated". The phrase is written in its abbreviated form at the end of a mathematical proof or philosophical argument to signify that the last statement deduced was the one to be demonstrated; the...
External links
- Squeeze Theorem by Bruce Atwood (Beloit College) after work by, Selwyn Hollis (Armstrong Atlantic State University), the Wolfram Demonstrations Project
The Wolfram Demonstrations Project is developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
.