Squeeze Theorem Proof

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Squeeze Theorem Proof

In this video, I prove the squeeze theorem, which is a very classical theorem that allows us to find limits of sequences.

Other examples of limits can be seen in the playlist below.

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I have always wanted to see this proof!! This theorem is fabulous and the proof is superb!!! Thank u dr.P love u

Kdd

In Russia we call it 'two policemen theorem' (like a two policemen (a_n, b_n) and a drunk between them (s_n))

pufitus

Un excelente video. Este teorema en particular es utilizado frecuentemente en teoría de muestras grandes (Large sample theory) en estadística. Gracias por este video, muy útil para nosotros.

MrCigarro

*Squeeze Theorem Proof*
*Dr Peyam*
_Class notes_

Published on Jun 30, 2020
_Squeeze Theorem Proof_
_In this video, I prove the squeeze theorem, which is a very classical theorem that allows us to find limits of sequences._
_Other examples of limits can be seen in the playlist below._

Definition of a Limit:

Check out my Sequences Playlist:

The *squeeze theorem*, also known as the *pinching theorem*, the *sandwich theorem*, the *sandwich rule*, the *squeeze lemma* . In Italy, the theorem is also known as *theorem of carabinieri* .
In many languages (e.g. French, German, Italian, Hungarian 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.

[ SQUEEZE THEOREM PROOF ]
*CLAIM* if aₙ, sₙ, bₙ are sequences
such that *aₙ ⩽ sₙ ⩽ bₙ*

And Limₙ ⃯ 𐨍 aₙ = s
and Limₙ ⃯ 𐨍 bₙ = s
then Limₙ ⃯ 𐨍 sₙ = s

Let ε > 0 be given
We know:
1) There is a N₁ such that
if N > N₁
then |aₙ - s| < ε
⇒ -ε < aₙ - s < ε
⇒ *s - ε < aₙ < s + ε*
2) There is a N₂ such that
if N > N₂
then |bₙ - s| < ε
⇒ -ε < bₙ - s < ε
⇒ *s - ε < bₙ < s + ε*

Let N = MAX{N₁, N₂}, then,
if n > N
then
-ε < aₙ - s ⩽ sₙ - s ⩽bₙ < s + ε
⇒ -ε < sₙ - s < ε
⇒ *|sₙ - s| < ε* ✔️
_we're done!_

HENCE Limₙ ⃯ 𐨍 sₙ = s ◾
( _End of proof_ ◾= Q.E.D.
_Quod Erat Demonstrandum_
Which was to be demonstrated)

alexdemoura

I have a doubt
Can { lim as n goes to infinity of (sn+tn) } equal { lim as n goes to infinity of sn} + { lim as n goes to infinity of tn} IF sn and tn diverge??

sanjeevkumardhiman

In German it has the ugly name "Einschnürungssatz", but i like the name Sandwitch Theorem.

cerwe

in italy this is the "carabineros" theorem :)

russellkane

Could this theorem be true if a_n>s_n or s_n>b_n for some natural numbers n such that they form a finite set?

luisibalaz

When I first took calculus in the late 1970s, this was called the pinching theorem in books.

historybuff

Cool vid Dr Peyam. Doing this one for homework made me feel like I was actually getting the hang of this analysis thing haha

sebmata

in israel, the roughly translated name of this theorem is "the sandwich theorem"

amitir

You are amazing, keep doing what you do!!

martyalden

I love this channel. Dr. Peyam is just incredible. I have learned so much from him and love math just like him. I feel so inspired that I wanted to make youtube videos about my journey through mathematics, which is about to go to a new level as I start high school. I would love some support!

leoyang.

J'ai entendu ''voilà ''.

federicodesoya

Squeeze me with your hot theorem! Lol 😂

But seriously you rock

NotoriousSRG

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This is a very famous limit