Real Analysis 7 | Cauchy Sequences and Completeness

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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
00:14 Convergent sequences
00:54 Different property of a sequence
02:09 Definition Cauchy sequence
02:16 Connection to convergent sequences
03:14 Dedekind completeness
03:47 Sketch of proof
07:35 Application for monotonic sequences
08:44 Credits

#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)

[lucvansprang]

imagine taking real analysis before this series, i love you man.

[michaeltamajong]

If it is not an application, this man has the most organized handwriting I've ever seen!

[qiaohuizhou]

1:31 idea of Cauchy sequence, the sequence eventually lay arbitrarily close to each other
2:24 fact: Cauchy sequence is equivalent to convergent sequence when dealing with reals
3:15 dedekind completeness and the properties for subset of real numbers
3:51 proof ( read again )
7:40 criterion for convergence of sequence

[Algebraictivist]

Thank you for the amazing explanation, but how do i get the (1/2)^n-1*|b_1-a_1| part at 6:58

[theklausjesper]

Insane my guy you are really saving me

[NotFound-bgsr]

This is interesting coming from your presumably newer videos on introductory math concepts, where you motivate convergent sequences using Cauchy sequences, the other way around. I found this other approach helpful too

[yaoweizhang]

Dear
The Bright Side of Mathematics,
Last video on your list is still set as private video. I guess it is meant to be a video talking about sup and inf.
Regards.

[monadic_monastic]

I had no clue this was also called the join btw, I've been learning some geometric algebra and wonder if there's a relation to its non-linear join and meet operators.

[xoppa]

Dear brightside, excellent video. One issue i am doubtful about.
The definition of a cauchy sequence states that for any epsilon > 0 there is an N such that for any m, n >= N (note the comma between m and n, not just m > n) then |a_n - a_m| < epsilon.
However in your proof at 6:33 you showed that m > n . Do you have to do a seperate proof for n > m ?
In the expression |b_n - b_m| it looks like b_m is varying while b_n is fixed, but you could also vary b_n and fix b_m, or does it make no difference because of absolute value, i.e. without loss of generality.

[PunmasterSTP]

Cauchy? More like "Cool and catchy!" These videos are very nice; thank you for making and sharing them.

[enriquecorimayo]

I have a question... in the last part of the video, shouldn't be "the existence of the infimum of this set?" instead of supremun ...minute 8:15. Thanks

[chair]

That convergence application was not trivial and I wish you would have gone over proving it.

[ArittraMalhotra]

In my class the professor used Dedekind Completeness as an axiom.
But you showed a proof for it so what is correct ?
Also thank you for these amazing videos, the small diagrams help a lot in visualization !

[Leslie.Green_CEng_MIEE]

At 5:36 and 6:00 there is an typo with the iteration values in the *case (2)* parts.
(I am using an underscore for the subscripts)


In the *case (2)* assignments, when we don’t have a closer upper bound, it is written as
a_2 := x
and
a_(n+1) := x

These should read
*a_2 := c_1*
and
*a_(n+1) := c_n*

*Discussion* x is a value which exists, showing that c_1 is not an upper bound. Agreed. However, x is not a specific iterated value. Certainly there is a value x > c_1, we just don’t know what x is.

If you like, the iteration has left values (a’s) and right values (b’s). We pick which ‘side’ is changed to progress the iteration. The calculated c value goes either to the left or to the right.

[michaelnicodemus]

I don't understand why you define a_n+1 as x in the second case. Why is it not c_n?... You say there exists an element x in M such that x > c_n, but how do you pick an x when there are multiple choices (possibly infinitely many)?

[filmmyduniya-mfhq]

Thank you sir for this wonderful lecture but i did not understand why the limit is SupM and also in case of (1+1/n)^n the limit is e how do we now this by conv. Criteria

[Study-lxlt]

Thanks for the quality content, I'm just wondering what can I do after watching the videos to improve?

[maiSenpaiDaisuki]

7:01 hello, I don’t understand why did you put (1/2)^n-1, I think this power will be increasing instead of decreasing. I think it will be more reasonable to put n+1 instead, so that the denominator will get larger and larger, then it will actually be decreasing. Can someone explain?

[elidoz]

there is one thing I don't understand
for example with the sequence where a_n=1/1+1/2+1/3+...+1/n
the terms of the sequence get arbitrarily close to each other, but at the limit it is infinity and doesn't converge

[gingervacation]

the proof of completeness reminds me of binary search