Real Analysis | Compact sets, supremums, and infimums.

We prove that compact sets of real numbers contain their least upper bounds and greatest lower bounds.

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[goodplacetostop]

6:03 Good place to stop before showing the set of integers which are both even and odd.

[richardbloemenkamp]

The Cantor set is a closed and bounded subset of the reals. Therefore it is compact. Another definition of compactness is that every open cover of a compact set contains a finite sub cover. We can define a open subset covering the Cantor set that is not finite. One such open cover is the infinite set of open intervals where each open interval is 10% wider than and each closed interval that is part of the Cantor set and each open interval is centered at the center of the corresponding closed interval. This infinite open cover does not contain a finite sub-cover that is also a cover of the Cantor set. Therefore the Cantor set is not compact.

Should I conclude that compactness is the Cantor set?

[사기꾼진우야내가죽여]

Using completeness axiom to obtain the supremum and showing that it is contained in the set is exactly the same process as proving maximum value theorem.

[edwardjcoad]

Always struggle with this type of maths without there being worked examples. Can we have a video which takes a worked scenario of meaning which uses this analysis to show benefit.

[tomkerruish]

Well done as always, but are you at some point do point-set topology? I know you're just doing real analysis now, but "every open cover has a finite subcover" makes things so much easier, like proving that the image of a compact set under a continuous function is again compact.

[killdm]

after there s bolzano-weierstrass theorem?

[noahtaul]

Why do these videos keep stretching out to the 17 minute mark? Are you posting them and then not checking them?

[dzakytamir]

Please make a video about examples from all theorem in this playlist

[CM_France]

Hi,

I ask the same question as many of us: why do use 50 or 100% more time recording for nothing? Think about the planet and the humanity.