Thank you for the lectures, Professor. It has been a pleasure!

Fantastic lecture!

Thank you so much

Love watching this first thing in the morning with a cup of coffee. ~smiles

Sup

I am in the set of comments

Prof made a mistake on the board when he wrote down the "definition" of the subset....he wrote A is a subset of B if x in A then x is in B. You need a universal quantifier right in front to indicate that you are talking about every element in A, is also in B, not just a particular x, or some x.

homomorphism of Z into Q (trivial kernel）

I can't see anything. Recorded on a potato

1:05:25 A GAT CAT 😾🔫😤😤

45:00 His example is not very good. In E=X, *every* point of E is an interior point of E. Just take any neighborhood of p in E. I think he got mixed up.

Most people would get it easily what real analysis is if we change the name to be "real numbers overanalysis"

Every one knows about the first YT video about the elephants....is this the 2nd YT video?

Thanks a lot Mr. Francis Su ❤

Forgive the poor production and just keep watching. This guy is the one. Sometimes I think he may be trying to tell us something with the fuzzy resolution and strange aspect ratio.

Any group theory lecture might be helpful.. please..

40:50 if you don't have to check the condition thst the empty set is bounded by 2, why does that mean it is true that it is bounded by 2 (vacuously true).

1:05:20 what is the property in multiplication and addition that preserves that order.

This guy really motivates what's important in proofs

Why does he assume the integer 5 is the rational number 5/1 at 20:45. OK, I think he justifies to some degree in the next minute by saying that an integer n and an integer m, expressed as n/1 and m/1 respectively will belong to equivalence classes in Q satisfying the usual multiplication and addition rules. I think that's what he does?

Nice lectures. Why is equivalence defined as having these three properties. What's the intuition behind choosing just these three things to describe this notion of equivalence

Maybe I missed something, but it wasn't proven that changing elements of any set, countable or not, would make it uncountable. e.g., Let A = {all letters of the alphabet}. Then this should be countable, but if I take out S and put in -Z, then it's uncountable. So we can do this for every set. So, to reiterate, prove how changing the elements of a countable set wouldn't change it's countability status. Seems like it would since if we're allowed to change elements at random, no set is safe from this violation.

forgetting my dots & crosses, I don't remember why there's a negative bd, then a positive bc for xy. I remember i, j, k, but this isn't that.

Why wouldn't the contradiction from 52:38 be the same problem for y? Change alpha to y, x to alpha, then y isn't an ub for A either. how is this wrong? So using precisely the same proof method, y - alpha < m*alpha for some other n ∈ N. So, y < (m*alpha +1)alpha, so y is NOT an ub for A. I find things like this frequently in RA & topology, which is why I don't get it.

Jeez, I really wanted to stop him from calling (0,1) "i"

17:30 It's hard to figure out at first (assuming I got this right) that a isn't the same exact value as a', but that the entire fraction of which a is a part is equivalent to the fraction involving a'. With so many new concepts flying around, it can be like trying to construct a house from a tornado.

Day to Celebrate: I actually understood what you wrote on the board around 42:00, gonna hit the all you can eat buffet.😃

gonna miss you Professor Francis Su 😢

bro sounds like a half-life scientists

This was highly recommended but the video quality is awful. I can barely read the blackboard - sometimes.

integers dont really occur naturally , we have negative integers that dont really have natural representation

1:08:30 set e = a/b and choose n sufficiently large that n! a/b is a natural number.

I lost my glasses the other day so these are too blurry for me to see, but they sound like great lectures!

Thank you!

2:29 I think the stdudent does have a point there, In the real world everything is disrete and wheunting atomsn you zoom far enough you'll essentially be a collection of atoms. Real numbers and Continuity are just an approximation of the very dense collection of atoms as continous matter.

The professor keeps saying ''see the book''. Can one of you guys tell me the name of the book? thank you in advance.

54:00 Relative open sets

0:42 Recap 3:15 New representation of compactness: A set is compact iff every collection of closed subsets of it Satisfies FIP 12:15 Cantor sets. 21:27 Another representation of a cantor set. 27:39 Cantor sets are uncountable. 29:53 Why do cantor sets have no interior points? 30:17 Why every point in Cantor sets is a limit point? 32:10 If the number has finitely many digits (0.02000...), then is it a limit point? 34:43 Cantor sets are disconnected. 35:47 Cantor sets are length zero: for every desired epsilon, it can be covered with intervals of total length less than epsilon 38:22 Base/Basis: what you can build everything else from | Every open set is the union of some base elements 41:27 Base is important to determine size of a set 41:50 Compact metric space iff has a countable base 44:01 Connected sets 53:25 another definition of connected sets

Recap 1:03 Closed Intervals are compact 6:27 Questions about proof 22:08 Closed and bounded sets are compact in R^n (Heine-Borel) 25:40 Question about proof 32:31 Examples of closed and bounded sets which are not compact 36:30 A set is compact iff every infinite subset of it has a limit point 44:43 Corollary (Bolzano-Weierstrass) 1:00:20 Cantor's finite intersection property 1:03:03

being used 13 years later in my winter semester lol

Bravo!

amazing lectures! Although, the video quality is very bad :( This lecture specifically seemed kind of quiet too.

51:00

thank you very much teacher

The point about "if"s in definitions was really good.

A very insightful lecture! Thank you so much!)

I just understand the Dedekind sets today. Thanks to u prof

Thanks a lot teacher.

I don't really understand the induction step of the transfinite induction example. Can someone explain how we verify that K_α satisfies the smiley conditions?

Thank you professor.