In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) 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 , 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. 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. Statement The squeeze theorem is formally stated as follows. The functions and are said to be lower and upper bounds (respectively) of . Here, is not required to lie in the interior of . Indeed, if is an endpoint of , then the above limits are left- or right-hand limits. A similar statement holds for infinite intervals: for example, if , then the conclusion holds, taking the limits as . This theorem is also valid for sequences. Let be two sequences converging to , and a sequence. If we have , then also converges to . Proof According to the above hypotheses we have, taking the limit inferior and superior: so all the inequalities are indeed equalities, and the thesis immediately follows. A direct proof, using the -definition of limit, would be to prove that for all real there exists a real such that for all with , we have . Symbolically, As means that and means that then we have We can choose . Then, if , combining () and (), we have which completes the proof. Q.E.D The proof for sequences is very similar, using the -definition of the limit of a sequence. Examples First example The limit cannot be determined through the limit law because does not exist. However, by the definition of the sine function, It follows that Since , by the squeeze theorem, must also be 0. Second example Probably the best-known