Real Analysis 5 | Sandwich Theorem

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:10 Limit theorems
00:50 Monotonicity of the limit
01:45 Sandwich Theorem
02:33 Proof of the Sandwich Theorem
05:30 Example
07:48 Outro

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[desrucca]

4:43

I think its better to explain why (d_n + a_n) converges to a

Its because
lim{n→∞} (d_n + a_n)
= lim{n→∞} (d_n) + lim{n→∞} (a_n)
= 0 + a
= a



I got confused by the form :
lim C_n = |d_n **+** a_n| < elipson
with
|c_n - a_n| < elipson
in minutes because im trying to slove it based on the original form :

**a is the limit of a_n** if **|a_n - a|<elipson**

[lucaug]

Really enjoying this series! Good content as always!

[Beancp]

I'm loving your videos! It has been a joy to watch them in my free time. I'll definitely be supporting your work as soon as my financial situation allows me.

Now a question: does Cn have to be convergent by hypothesis? It seems to me that the proof guarantees such a "sandwiched" Cn is convergent

[sereya]

For the sandwich thm isn't it enough for the inequality to hold for n>N for some N? We are dealing with limits here so I assume it should be enough

[Lumell]

Great job! You made it easy and straightforward.

[Leslie.Green_CEng_MIEE]

As a cross-reference to other authors, *"Sandwich Theorem"* is also known as *"Squeeze Theorem"* .

Remarkably good content.😊

[ahmedamr]

Great video once again! One question:
Is there a reason why you flipped the order of lim a_n and lim b_n at 2:53? Shouldn't it be the b_n - a_n?

[jchan]

Hi Professor, monotonicity property of the limit, I am not clear why greater is not correct and must be greater or equal?

[hugoreyna]

Hi, good job. Thanks for sharing.
I have a question, what's your editor video?

[hetsmiecht]

I was having some difficulties with one of the quiz questions on your website. The task is to use the sandwich theorem to find the limit of 1/sqrt(n). I know that 0 is a lowerbound for this function, but I can't find a suitable upperbound which has 0 as its limit. Could someone help by explaining how one would go about finding an upperbound for this function?

[franciszekwieczorek]

I understand everything up to 7:17 but then why is 1/(sqrt(n^2 + 1) + n) <= 1/n? Do we choose arbitrary epsilon here?

[PunmasterSTP]

Sandwich theorem? More like "I want to be near them"...your videos that is! Thanks again for making these amazing series (no pun intended).

[tayebtchikou]

This theorem is very interesting because it can be used also to calculate the limits of some functions like sin(x)/x

[sinanakhostin]

In 2:46 you said that the limit (b_n - a_n) is zero. Why? I assume it can happen only in the case where the limits of "b_n" and "a_n" are the same. But here and in general case we do not know that. Or I am making a mistake ?

[sentralorigin]

7:12 how do you know C_n is always greater than or equal to 0?

[RuizhanGu]

where is the answer of this video's excercise.I couldn;t find any answer on the internet.

[VaheTildian]

At 2:55, why can you say it is positive or zero from the inequalities? I feel like I'm missing something obvious, but I'd rather ask

[notaweeb]

You're really saving my ass! Even if not, I'm really enjoying your lectures<3

[triton]

With 0 ≤ Cₙ ≤ 1/n, since Cₙ ≠ 0 for all n, would it be better to use strict inequalities instead?