Squeeze theorem

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of Carabinieri.Let I be an interval having the point a as a limit point. 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 haveLet ∑ n a n , ∑ n c n {displaystyle sum _{n}a_{n},sum _{n}c_{n}} be two convergent series. If ∃ N ∈ N {displaystyle exists Nin mathbb {N} } such that ∀ n > N , a n ⩽ b n ⩽ c n {displaystyle forall n>N,a_{n}leqslant b_{n}leqslant c_{n}} then ∑ n b n {displaystyle sum _{n}b_{n}} also converges. In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of Carabinieri. The squeeze theorem is used 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 Carl Friedrich Gauss. In many languages (e.g. French, German, Italian and Russian), the squeeze theorem is also known as the two policemen (and a drunk) theorem, or some variation thereof. The story is that if two policemen 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 policemen) the prisoner must also end up in the cell. The squeeze theorem is formally stated as follows. This theorem is also valid for sequences. Let ( a n ) , ( c n ) {displaystyle (a_{n}),(c_{n})} be two sequences converging to ℓ {displaystyle ell } , and ( b n ) {displaystyle (b_{n})} a sequence. If ∀ n ⩾ N , N ∈ N {displaystyle forall ngeqslant N,Nin mathbb {N} } we have a n ⩽ b n ⩽ c n {displaystyle a_{n}leqslant b_{n}leqslant c_{n}} , then ( b n ) {displaystyle (b_{n})} also converges to ℓ {displaystyle ell } . From the above hypotheses we have, taking the limit inferior and superior: