# Squeeze theorem

Squeeze theorem "Sandwich theorem" redirects here. For the result in measure theory, see Ham sandwich theorem. Illustration of the squeeze theorem When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.

In calculus, the squeeze theorem (also known as the sandwich theorem, among other names[a]) is a theorem regarding the limit of a function that is trapped between two other functions.

The squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Carl Friedrich Gauss.

In many languages (e.g. French, German, Italian, Hungarian and Russian), the squeeze theorem is also known as the two officers (and a drunk) theorem, or some variation thereof.[citation needed] The story is that if two police officers are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the officers) the prisoner must also end up in the cell.

Contents 1 Statement 1.1 Proof 2 Examples 2.1 First example 2.2 Second example 2.3 Third example 2.4 Fourth example 3 References 3.1 Notes 3.2 References 4 External links Statement The squeeze theorem is formally stated as follows.[1] Theorem — Let I be an interval containing the point a. Let g, f, 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 {displaystyle g(x)leq f(x)leq h(x)} and also suppose that {displaystyle lim _{xto a}g(x)=lim _{xto a}h(x)=L.} Then {displaystyle lim _{xto a}f(x)=L.} The functions {textstyle g} and {textstyle h} are said to be lower and upper bounds (respectively) of {textstyle f} . Here, {textstyle a} is not required to lie in the interior of {textstyle I} . Indeed, if {textstyle a} is an endpoint of {textstyle I} , then the above limits are left- or right-hand limits. A similar statement holds for infinite intervals: for example, if {textstyle I=(0,infty )} , then the conclusion holds, taking the limits as {textstyle xto infty } .

This theorem is also valid for sequences. Let {displaystyle (a_{n}),(c_{n})} be two sequences converging to {displaystyle ell } , and {displaystyle (b_{n})} a sequence. If {displaystyle forall ngeq N,Nin mathbb {N} } we have {displaystyle a_{n}leq b_{n}leq c_{n}} , then {displaystyle (b_{n})} also converges to {displaystyle ell } .

Proof According to the above hypotheses we have, taking the limit inferior and superior: {displaystyle L=lim _{xto a}g(x)leq liminf _{xto a}f(x)leq limsup _{xto a}f(x)leq lim _{xto a}h(x)=L,} so all the inequalities are indeed equalities, and the thesis immediately follows.

A direct proof, using the {displaystyle (varepsilon ,delta )} -definition of limit, would be to prove that for all real {textstyle varepsilon >0} there exists a real {displaystyle delta >0} such that for all {displaystyle x} with {displaystyle |x-a|

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