Real Analysis | Sequences and the ε-N definition of convergence.

[@ashrayne4732]

You've just cleared up week 1 of Real Analysis in 8 minutes, thank you.

[@sarojpandeya9762]

Thank you for starting mathematical analysis.

[@drpkmath12345]

Pretty sure Michael has many threads ready. I am preparing for a full one year advanced undergrad real analysis video series as well. Ring theory would be interesting topic for making couple of separate videos for sure!

[@duckymomo7935]

I think so but he managed to film them ahead of time

[@drpkmath12345]

Mi Les yes I agree~

[@lucagirelli5223]

Yea I'm getting ready as well! This guy is so good at explaining things. There are also other people on YT that do good math videos but michael stands out for some reason, I'm also hoping for him to do olympiad problems because I want to become good at them and I always seem to learn something new everytime so yea, this guy is a monster

[@matrixmash9907]

Thank you so much for the real analysis playlist. This is immensely helpful, a lot of my classmates and I are struggling to understand the concepts fully as our textbook essentially blows through the definitions and propositions like their either the most trivial results or, for "brevity", leaves a lot of the back-end material out. Definitely shared this playlist.

[@cloriscloris5491]

Omg, this is so clear！I wish u were my professor.Thx for saving my midterm, Sir.

[@superorangebody]

Dude this is so good. I'm reviewing for a final exam in my intro to analysis class and your videos really help!

[@gilbertchan2696]

thank you, you've single handedly saved me from my real analysis subject

[@languageoflogic224]

Most proofs I’ve seen start with “let’s assume..“, but math is a science requiring facts to be focused on to avoid assumptions. This guy starts with “given” and he goes through the proper steps. Subscribed.

[@noahtaul]

Wow, how many threads do you have going on at once? Sequences, group representations, differential topology, contest problems, maybe ring theory...

[@FabianAguilar-u8v]

Thank you. This was a great review before my class.

[@valeriereid2337]

Very well explained. Thanks for your contribution to mathematics education.

[@AilinZeng-s8t]

wow this is amazingly clear thank you so much. my professor doesn't really explain anything i always get so confused listening to his classes

[@tomatrix7525]

Best analysis stuff I’ve ever seen

[@mimodite]

Maybe this is semantics but isnt it wrong to say that the "band" (strip which is epsilon distance from L) gets smaller when n gets larger? Isnt it other way around? In the sense that if we take smaller epsilon, n that is sufficiently big probably gets larger?

[@conchoprimo]

I think you are right.

[@ayeshasiddiqua9018]

Really found helpful...now I literally understood the concept... thankyou!

[@chaoticlife5569]

@michael penn I'm using your videos to review my real analysis course. I wish in my course, they mentioned you choose the value of N in your "scratch work". It was so hard for me to wrap my head around that idea. At the time it felt like going on a loop and doing the proof twice.

[@ichkaodko7020]

awesome demonstration.

[@Adam-wm4ys]

King!! You dropped your crown

[@thelogicone185]

Please, can you help me? I have a doubt. Seems like everything is explained: n>=N and |an-L|0 But and L? L is not explained. What L needs to be?

[@Will-nf9gf]

L is supposed to represent the supposed Limit of convergence

[@schweinmachtbree1013]

Any reason you prefer the brace notation {a_n}_{n=1}^{\infty} instead of the bracket notation (a_n)_{n=1}^{\infty}? I particularly don't agree with your notation {a_1, a_2, a_3, ...} (top left of the board), since that is the notation for a set, so there is an issue with repeated elements in the sequence.

[@sajidbinmahamud4605]

Thanks Michael!

[@TheSpark183]

I love your videos! Keep it up! You are saving lives :D

[@bobbysokhi7296]

Extremely Helpful. Cheers !

[@nabilalhoussein7877]

At the end of the video, why did you add 1 and subtract 1. We could also add 2 and subtract 2, does that mean that the limit is 2? I don't think so.

[@williamroywhite]

That would contradict the scratch work. Remember what he is trying to prove is that |(1-1/n) -L| < ε => |1/n-1 - 1| < ε

[@jadeconjusta1449]

This is so helpful. Thank you!

[@abhijitharakali]

Mystery of dislikes continues! I witness many videos of superb quality, getting dislikes. What's the deal with you people? Mike is doing a favor for strangers by providing short tutorials on useful material. At least, abstain from watching any of his videos if you, for some mysterious reasons, not like the content. Why watch the video and hit the dislike button?

[@ayoubnouni224]

Or you can take N =[1/sqrt(epsilon)]+1 [***] means the floor function

[@ayoubnouni224]

Angel Mendez-Rivera I don’t think so

[@claireelizabeth2057]

So happy for some real analysis content! Discussion of Riemann-integrable functions would be much appreciated if you were into suggestions.

[@bilalabbad7954]

4:24 one over epsilon .it can not be always natural number

[@williamsteph2104]

This is really great thanks!

[@duckymomo7935]

are you gonna do topology? (open sets, etc)

[@math-ch9zk]

Really very helpful... Thank you

[@tgx3529]

And what about this definicion? An infinite sequence is defined on a subset of natural numbers, its cardinality is the same , its alef 0??

[@si_monster7365]

Why do you call the stuff with epsilons N?

[@guillaumest-onge613]

I might have missed it in another video, but I have always wondered : why is it important to reverse the "scratch work" proof in the end?

[@stefanplusplus917]

The scratch work is just the process of creating the number required for your proof. It is not an actual part of your proof. The proof is to show that the number you found satisfies the conditions of your statement.

[@williamroywhite]

@@stefanplusplus917 but surely if you have done the scratch work correctly then since the proof is just the reverse of the scratch work the proof MUST work? So then why is the scratch work not sufficient for the proof?

[@사기꾼진우야내가죽여]

For each epsilon greater than 0, the set of all natural numbers n such that nth term of the sequence is not in epsilon neighborhood of x is a finite set. Equivalent statement of definition of convergence of a sequence

[@TheSGOGS]

dope video thanks really helps

[@jvo807]

Brilliant video damnit💯

[@strider_1237]

Thank you so much!

[@itsssssssu2086]

Thanku ❤️

[@muhamadasepridwan8195]

Thank You

[@urty4395]

you're a god!

[@roberttelarket4934]

There is no reason for 4 thumbs down!

[@predatorcorp3366]

How we can guarantee that as N get bigger then epsilon get smaller

[@conchoprimo]

By the power of wishful thinking

[@samisiddiqi5411]

Excellent video but the only issue is the ads are in inappropriate places. This is distracting and makes me lose focus on the proof. I know this is not your fault but KZbin itself. However if possible, can you please put the advertisements in places where it isn't the middle of a proof?

[@generalmathematics8499]

I missed that back flip ;-)

[@quant-prep2843]

the god

[@emir2750]

Öz abim ya

[@azulpurpura1230]

I think I love u

[@coreymonsta7505]

3 dislikes???