Real Analysis | The Heine-Borel Theorem

[@MichaelPennMath]

I just woke up to notice this video cuts off in the last step. Sorry everyone! My files have been strangely exporting recently... I will make a video with the last step and post it unlisted later today -- linked at the end of this video.

[@MichaelPennMath]

Here is the new video: kzbin.info/www/bejne/pmrRfKinr5t3b9k

[@goodplacetostop2973]

17:54 “So this is...” not a good place to stop

[@CDChester]

GOT EM

[@numcrun]

lol

[@Jim-be8sj]

One of two things happened here: 1) Cauchy sequences do not necessarily converge in the KZbin space. 2) Gödel's famous theorem was being demonstrated in some way.

[@bencheesecake]

One minor note is that we must show that S (from 12:08) is nonempty before determining the existence of its supremum. This is done by taking some U_i containing a, and finding some x_i such that [a,x_i] is contained within U_i, then this single U_i provides a finite (containing only one element) subcover of [a,x_i], so x_i is in S. For those of you preparing for exams, this is the kind of detail that I have gotten dinged on in my oral exams.

[@mohithraju2629]

The rest is left as an exercise to the viewer.

[@raphaelkelly861]

The rest of the solution is trivial and left as an elementary exercise

[@ian731]

@@raphaelkelly861 a tensor is something that transforms like a tensor.

[@get2113]

Nice treatment of tougher proof, but there is an editing problem near end .

[@philperfect8800]

Lemma: the proof look like an induction. But don't you have to prove first that S is not empty. In that way, could we say that [a,a] is obvious part of S.

[@joaomegazen]

You are saving me...my master exam is next month and it's all real analysis! Thank you for your classes!

[@PowerhouseCell]

*Imagine watching the whole video waiting to see how it ends 😢*

[@aristo7051]

Can someone help explain why c < b is a contradiction of s @16:15 thank you

[@alegal695]

Case c' != b. Since c' = min{ s + eps, b } --> c' = s + eps --> s = c' - eps --> s < c' which is in S --> s is not an upper bound of S --> s != sup S, which contradict that s = sup S.

[@AmanGupta-sj1rx]

Video just stopped 😩😩 Eagerly waiting for the ending 😩😩

[@tomkerruish2982]

Well, this answers my question from yesterday. Apparently I just need patience.

[@sitienlieng]

At 10:26, we contradicted to the hypothesis by showing y must be in K, but I could not see how y must be in K. In fact, we actually contradicted to the hypothesis that some terms of {y sub n} is not in K, and it makes more sense to me. Is there a mistake?

[@heewahhin7470]

We started off with a K that satisfies the property that every open cover of it has a finite subcover. We also assumed that the K has a limit point 'y'. If K does not have a limit point then K is simply closed. From the proof, we know that if such K exists, i.e. satisfying the two conditions then y not being an element of K is not possible. Therefore, if this K exists then we must have all of its limit points to be in K, which makes K a closed set.

[@CM63_France]

"Ok, great". I think we'll call you the Hitchcock of the maths :)

[@KurdaHussein]

where can I find that proof for Rⁿ ?

[@valeriobertoncello1809]

I have only one concern: when we define epsilon using the minimum function, aren't we assuming that a minimum exists? Why should this be true in general?

[@AP0PT0SIS]

The set is finite by the finite subcover condition, so finite sets always admit a minimum element by the well ordering principle.

[@pandas896]

I couldn't understand anything. And doesn't makes sense , how is it useful?? Can someone explain in what are the uses of these things? Any use in physics?

[@DarGViD]

That's basically just a part of the common language of advanced analysis. (Physics loves it)

[@duckymomo7935]

The result isn’t applied directly but is important to use in physics sure Complex numbers is a nice field with compactness property A notion of finiteness is nice Many theorems use compactness (see compact support)

[@caladbolg8666]

My favorite theorem

[@Ahmed2-hz7yz]

At 9:58, Why is it true that yN doesn't belong to any of U sub xi?

[@fahadqureshi1226]

By definition, the limit (introduced around 7:18) must be within an infinitely small distance from the terms in the sequence. However, U sub xi caps this distance at 1/2 min {(x1-y), ...., (xn-y)}.

[@philperfect8800]

I do not understand why c'= min(s+epsi,b). The sets of finite cover Ui are not constraint to be in S, nor in [a,b], so the upper bound of the cover could be anything bigger than c. Your demo is a special case wher [a,c] = Union of Uin.

[@carl3260]

The aim is to prove finite cover of [a,b], by proving a finite cover of [a, s+e] where either s+e=b and so the required finite cover exists (that it may cover more than [a,b] is irrelevant)

[@user-A168]

Good

[@GrandMoffTarkinsTeaDispenser]

KZbin: *This is a good place to stop*

[@CDChester]

ahhh more analysis

[@Wultuswuffel]

youtube thinks Im smart enough to understand any of this

[@lyjv]

this looks so useless