Real Analysis Ep 2: Bounds, sup and inf

Episode 2 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. I had my hand over the microphone for a lot of it, so the sound is messed up. Sorry.

This episode is some more random introductory stuff, our first theorem with an epsilon, and definitions of the sup and inf.

[fatmacdacat]

Hello sir, I just wanted to reach out and say thank you. Your amazing lectures are the only reason I am passing my class this term. I can't thank you enough.

[christopherrosson]

This is some of the best lectures i have found in youtube. Well done and thank you

[rikyikhwan]

Very intuitive, your lecture is amazing !

[Gunbudder]

This course is really interesting because when i was getting my engineering degree, i had to take a 4 class calculus series that included what you've lectured about (at least so far). I did SO MANY formal limits before we were allowed to "cheat" and do the shortcuts. and god help you if you ever said a limit was "equal" to something.

The exact wording my professor used was "arbitrary closeness" instead of "really really small." I was also not required to study differential eq or multivariable for my degree though, which is kind of insane. i REALLY regret not learning those two subjects in school because i had to learn them anyway on the job! It was pretty awkward the first time one of the senior engineers assumed i knew all about solving differential equations... I am a real time software engineer though, so i'm not expected to know the ins and outs of the purely physics or math side. i'm more on the logic side (i do know what a k-map is!)

[bobbun]

I would love to have had a resource like this to view in preparation for taking advanced calculus. Having said that, though, I feel it's almost cheaty to assume the real numbers rather than defining a linearly ordered set, describing inf and sup in those terms, then constructing the real numbers via Dedekind cuts. I guess I had it the hard way! We got about 2/3 of the way through the Rudin text in two semesters.

[ms__levitating]

Sir Thank You So Much Again for this Lecture🥺💖

[teddi_tqt]

Hi professor,

I was reading through the book along the course, and it seems that you left out the part where we compare max vs sup (min vs inf)
p.16 of Abbott's book (definition 1.3.4).

It can be important because I remember we will use max{N_1, N_2} in some proofs later on.

Like always, it is a wonderful lecture. Thank you so much.
It's a bit ironic that the pandemic pushed forth this series of lectures available on the internet.

[narattamchakraborty]

Good explanation .. I'm gonna watch all your videos professor

[andreajirasek]

thank you for being the reason that i am gonna pass my midterm

[sakr]

Thank you SO MUCH for these amazing lectures.

[christianflores]

Hi Professor, I love these notes. I'm unsure if you still monitor comments, but a question I have is how this material compares to Rudin's book? My school uses that textbook and I thought of using your lecture series as prep material. Would you say that is sufficient?

Also, what are some ways you recommend to work on proof writing if someone has never been exposed to it before?

[bansishah]

I surely will be following and watching the rest of your lectures on real analysis. Could I possibly reach out to you for any doubts or questions I have ?

[shanmugasundaram]

Hi Professor, I very much like your lectures.

The class webpage is not accessible though, could you please check the link you have given in the description. I would like to attempt the course assignments as well.

Thanks.

[panchamiop]

thank you professor for the amazing session.actually i could not open the link given in the description

[alexandergutierrez]

Hi Prof. Staecker, thanks for posting these videos! It's been awhile since I was in uni, and I never took real analysis. I was wondering, for the second proof you do showing a = b, does the following work? I had paused the video and tried to work it out on my own before seeing your proof.

Assume a <> b. Then |a-b| > 0, and let |a-b| = c.
Since epsilon and c are both elements in the set of reals > 0, there exists a value of epsilon such that epsilon = c = |a-b|
Thus, there exists at least one number where |a-b| = epsilon, thereby contradicting the assumption that all values of |a-b| must be less than epsilon. Therefore, a must equal b.

Does that work? It's been a really long time since I've written a proof (almost a decade) and I wasn't particularly good with them at the time so I'm sure my logic is sloppy in there somewhere.

Really enjoying the videos and again, thanks for posting them to Youtube!

[Kudravets-Diana]

If i have a positive sequence {bn}. So why liminf(1/bn)= 1/limsup(bn) ..
Some one told me to just use Sequential Continuity .I do not really understand how it connects..:/

[EmilioGarcia_]

Profesor is it possible to have access to your lectures notes?

[stumbling]

Oh, I was looking for surreal analysis.

[igorfritz]

what intuitively is epsilon greater than 0 mean

[claymusic]

11:45 isn't E>0, |a-b| < E the same thing as |a-b| = 0.

Edit: I spoke too soon. We're trying to think of |a-b| as the "smallest positive number" instead of 0