In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. The theorem asserts that if two functions approach the same limit at a point, and if a third function is "squeezed" ("pinched", "sandwiched") between those functions, then the third function also approaches that limit at that point. Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

The squeeze theorem is a technical result which is very important in proofs in calculus and mathematical analysis. 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 mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss. Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... Archimedes (Greek: ; c. ... Eudoxus was the name of two ancient Greeks: Eudoxus of Cnidus (c. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... Carl Friedrich Gauss was a German mathematician and physicist. ...

In Italian and Russian, the squeeze theorem is also known as the two carabinieri theorem or two militsioner 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. Militsiya (Russian: мили́ция; Ukrainian: міліція; literally Militia) was the generic name for the police in the Soviet Union and a few other Communist countries. ...

The sandwich/squeeze theorem has no relation to the ham sandwich theorem. The Ham sandwich theorem, also known as the Stone-Tukey theorem in topology in mathematics, states that given n objects in n-dimensional space, it is possible to divide each one in half with a single (n − 1)-dimensional hyperplane. ...

Statement

The squeeze theorem is formally stated as follows.

Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have: In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Partial plot of a function f. ...

and also suppose that:

Then .

• The functions g(x) and h(x) are said to be lower and upper bounds (respectively) of f(x).
• Here a is not required to lie in the interior 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 → ∞.

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

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: In calculus, the squeeze theorem, (also known as the pinching theorem or sandwich theorem) is a theorem regarding the limit of a function. ...

Let ε > 0 be any fixed positive number. By the definition of the limit of a function, there is a δ > 0 such that: In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

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:

This completes the proof. Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

References

• Joseph M. Ling (2001) Examples on Limits of Functions: The Squeeze Theorem
• Dr. C. Sean Bohun The Squeeze Theorem