Real Analysis 2 | Sequences and Limits

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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.
This part 2 of the series with 64 videos. Here we define important notions like convergence and divergence.
00:00 Introduction
00:15 Definition of a sequence
01:50 Examples
06:06 Definition of convergence and divergence
08:55 Example and Archimedean property
11:56 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)
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[@ChrisSwaningAround]

Great stuff You do a great job of filling in those gaps the modern education can leave in a mathematics education 😁 Keep it up!

[@alexanderthegreat5352]

This is very helpful, but as a note I think it's more intuitive to explain the Archimedean property as : for any real number there is a natural number bigger than it and then use the corollary (there exist 1/n for all r>0 such that so and so) to prove love the good work

[@brightsideofmaths]

Yeah, this might be a matter taste for the Archimedean property. Thanks for the explanation there!

[@chilledvibes99]

yeah at uni we learn it this way :)

[@IsomerSoma]

I am watching your lectures as preparation for my 1st Semester. Thank you very much.

[@PunmasterSTP]

How'd your first semester turn out?

[@vickysharma2949]

Gefällt mir, wie du erklärst!😊. viel besser als Professoren an der Hochschule Baden Württemberg. Danke!

[@PunmasterSTP]

Sequences and limits? More like “Surely this playlist contains only hits!” Thanks again for making and sharing so many high-quality videos.

[@chin6796]

Thanks for uploading this series

[@researcher--]

You are an amazing teacher!

[@murilopalomosebilla2999]

Beautiful presentation!

[@realAhmedAbdElGhany]

i really like this channel, i really support this great effort i wish success for you in all the fields of your life !

[@brightsideofmaths]

Thank you so much 😀

[@Negs42]

@@brightsideofmaths Eine Anfrage bitte; Kannst du eine Serie über Differentialgeometrie machen? Danke

[@leyashalabi7209]

الله عليك يا احمد

[@dextreme1754]

Just started with this whole series I'm studying computer engineering and now I realised with your videos and explanation that Analysis is actually not that hard atleast on this level tho i had problems with even these simple concepts i guess it really depends on the teacher if the student can learn what they teach so I'm very grateful for you! Thank you!

[@bArda26]

Wow amazing and thank you for making these videos !

[@suryanarayanachebolu7870]

Hats off for your good Job.

[@declandunleavy8985]

Superb explanation as usual. You are a gifted communicator. One tiny piece of feedback - when watching videos on youtube there are many times when I want to pause the video to take in what is on the screen, like many other viewers I'm sure. This action always brings up the progress bar with its icons. If the text is written right to the very bottom of the screen it gets covered by the icons and can't be viewed on pause. Even after unpause it takes several seconds for the icons to disappear. An example occurs at the end of this video and I've noticed it on other vids too. Would be helpful in future vids if you left the bottom 5% of the screen blank. More an issue with youtube than your vids but unlikely they will change this anytime soon. Anyway - I hope you keep up the incredible work. Thanks!

[@brightsideofmaths]

Thank you very much! There is a really helpful feedback I also got before. I try to avoid writing at the bottom but sometimes I also want to show the stuff above at the same time and then I really want to use the whole screen. One possibility for you would be to use the pdf versions of the videos while watching the video. However, I try to be more careful in the next videos and try to avoid the bottom of the screen more often :)

[@kamildiril9544]

Thank you very much for an excellent explanation. I didn't understand the logic epsilon et N for 2 years. I understand them now.

[@brightsideofmaths]

You are welcome!

[@gianniskallionis3300]

as a greek, im proud of how u pronounce ε , good job !

[@user-kn6tm3zd1o]

Thank u❤ You’re Genius sir.

[@pratikkumar311]

Please upload more video on sequence and series.

[@laviekolchinsky9441]

Are you finished the series on constructing the basic number sets (R and C)? I've watched all the series you have except for these ones because I want to binge the full constructions and formalisms :)

[@brightsideofmaths]

Thank you! The construction of the real and the complex numbers is finished. However, I will add some videos about complex numbers soon but they are not need for the real analysis course here :)

[@gopinathan2095]

Great insight I got...

[@DrPeterkuah]

That is the best convergent definition I have ever seen /read.

[@brightsideofmaths]

Nice! And thanks for your support :)

[@inthebackwiththerabbish]

You are amazing Dankeschön

[@pratikkumar311]

Thanks

[@shaunsankar5397]

You're a great teacher, props!

[@dawardeka5404]

lots of Love from IIT Bombay

[@sadececns00]

Waowww you 100% cleansed all my confusions Thanks Sir!!!!!!!

[@brightsideofmaths]

Glad it helped! :) And thanks for the support!

[@sadececns00]

​@@brightsideofmaths Hello Sir! I'm watching the video again after 7 months I realized everything is much more clear now than it was 7 months ago. I have a question can we simply choose N=1/ε instead of showing for Archimedean property

[@brightsideofmaths]

@@sadececns00 The Archimedean property is given and we have to use it :)

[@sadececns00]

​@@brightsideofmaths Thanks for your reply : ] I follow another channel, too. And the teacher always do scratch work first to find delta (or N) then he chooses delta based on the scratch work. I haven't seen that he applies Archimedean Property- he just lets. He does this during the limits Epsilon- Delta proofs or N-delta M-delta proofs (for infinity).. Could you please tell me the reason why he lets Delta \ N

[@hz6612]

Thank you so so so much !!!

[@brockobama257]

I think I might pass real analysis

[@glennlehman9608]

I like the pace and content of your videos. Personally, I find them easier to follow than my text, so I appreciate the effort you put into the explanations. At the end of the video, you mention the Achimediam Property - I am not familiar with this. Do you have a video that explains the property?

[@brightsideofmaths]

Yes, you can find it here: tbsom.de/s/slm My Start Learning Mathematics series covers the construction of the real numbers. You don't need all the videos for understanding the Archimedian property. And thanks for liking me videos :)

[@glennlehman9608]

@@brightsideofmaths I think I might need all the videos ... Thanks Construction of the real numbers has been confusing me

[@shalvagang951]

hey you told that you will make a lecture series on algebraic geometry or differntial geometry so when you are making it

[@brightsideofmaths]

I try to do this on this year :)

[@user-cd9gw4qb4s]

I have a question. Why is 1 to the power of infinty indetermined?

[@brightsideofmaths]

Infinity is not a well-defined number for powers.

[@sivaramd5637]

So, a finite sequence never converges to any number "a"? Since, we can always choose 0 < epsilon < |a_n - a|.

[@brightsideofmaths]

A finite sequence is just a tupel and convergence makes not much sense there.

[@sivaramd5637]

@@brightsideofmaths Oh, okay. So, a sequence is infinite by definition (in real analysis)?

[@peki_ooooooo]

Good explanation.

[@brightsideofmaths]

Glad you think so!

[@aseelmathematics2778]

Great !!!!

[@abdeljebbarmnahaj7641]

Great job

[@shalvagang951]

hey can you also do a video on algebraic geometry and differential geometry

[@brightsideofmaths]

Yes, I can :)

[@shalvagang951]

@@brightsideofmaths so when you are planning to do that because I am very excited

[@maxpercer7119]

Fun challenge, negate formally the definition of convergence. For a bigger fun challenge, show that the sequence {-1, 1, -1, +1, ... } , given by the rule (-1)^n, does not converge. (Hint, it is not enough to show that the sequence does not converge to a specific number , e.g. 1 or -1, but we must show that it does not converge to any real number)

[@dodysumargo]

Does a limit pf a sequence mean the largest value that sequence has?

[@waraamir1487]

No for example if we take (-1)'n/n

[@vale_gamer5228]

Hello. I love the videos. Helping me a lot. Tho i still have a questions. Namely, what exactly is N(Not the N of natural numbers but just the capital N)? What does it represent???

[@brightsideofmaths]

Thanks! N represents a natural number, an index. So N could have the value 5, for example.

[@leochen4584]

at 10.28 why is it not 1/n >=1/N cos like it said n>=N

[@brightsideofmaths]

1/2 is bigger than 1/3 but 3 is bigger than 2.

[@cprt.d9471]

Wie nennt man die neighbourhood auf Deutsch? Wird es in der Mathematik ebenso mit Nachbarschaft übersetzt? :'D Und wie wird der archimedian properity übersetzt?

[@brightsideofmaths]

Neighbourhoood = Umgebung :)

[@ra1n0]

8:09 shouldn't the definition of convergence state that |an - a| with the latter a being the limit you picked (the red a)?

[@brightsideofmaths]

Yes, it should be a red a :)

[@tsunningwah3471]

bravo

[@intuitivelyrigorous]

Why |an-a| less than epsilon cant be |an-a| less than or equal to epsilon? Or it can be but both definition will prove the same convergence.(As there is always a number between two numbers so we will always have some other epsilon if not the epsilon which generates the case of equal to). Then is there any significance of choosing less than rather than less than equal to. If it cant then the case of equal to must be redundant, if so then how or the expression with equal to must not give proper idea of convergence.

[@brightsideofmaths]

< eps and ≤ eps give the same definition.

[@chunyunyang8999]

Hello, I am a student from mainland China, because of various reasons I can't pay, can you give me a PDF and channels to get your course

[@dodysumargo]

Does a limit of a sequence mean the largest value the sequence can give?

[@brightsideofmaths]

No!

[@dodysumargo]

@@brightsideofmaths alright then, so what does it mean exactly?

[@brightsideofmaths]

@@dodysumargo The value the sequence approaches for n to infinity.

[@dodysumargo]

@@brightsideofmaths Oh, alright then thanks for the info, im new to real analysis

[@brightsideofmaths]

@@dodysumargo You are welcome! You can join the community forum to discuss more problems :)

[@Rayglobster]

Why can’t you pronounce your R’s????

[@brightsideofmaths]

Genetic defect!

[@Rayglobster]

@@brightsideofmaths Sad!

[@brightsideofmaths]

There are worst things than that :)