Introductory Real Analysis, Lecture 7: Monotone Convergence, Bolzano-Weierstrass, Cauchy Sequences

(0:00) Exam 1 is in one week.
(1:01) There's a lot to discuss in this lecture to get ready for the test.
(1:44) Know definition of convergence of a sequence and be able to negate the definition and use that negated definition.
(2:05) Convergent sequences are bounded: know and be able to prove.
(3:06) Definition of "diverge to infinity" (be able to modify it to define what it would mean for a sequence to "diverge to minus infinity".
(8:31) Algebraic properties: know, be able to prove (with help on the trickier ones), and be able to use.
(14:06) The Squeeze Theorem: know, be able to prove, and be able to use.
(19:12) Convergent sequences whose terms are all in a closed interval will converge to a number in that closed interval.
(22:17) A monotone sequence converges iff it is bounded: know, be able to prove, be able to use.
(31:04) Subsequences and the Bolzano-Weierstrass Theorem (with an aside about the proof of the monotone convergence theorem): know and be able to use (proof is harder...would require help).
(36:00) Cauchy sequences and the Cauchy Convergence Criterion. Part of the proof is easy, part is hard.
(45:32) A look at some homework problems that could show up on the exam.

Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN.

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[harshvardhansingh]

The passion with which professor teaches is amazing. Thanks for posting these publicly. Best wishes from India.

[michaellewis]

I cannot seem to tell if being a graduated senior in high school I shouldn’t be able to be understanding these lectures, or you’re just that good a professor It seems the latter. The explications are so thorough. Rewatched just a few times.

[NilodeRoock]

Thanks to the indexes in many of the lectures, I use these lectures as a main reference, my first stop to simply remember or revise. So when I read a textbook, or a paper even, and some Real Analysis theorem pops up I seem to have forgotten ( e.g. Bolzano Weierstrass Theorem 31:21 : how could I have forgotten it! ) I go to YouTube / Bill Kinney. - My point being: this is more than a lecture series... it is a resource. Thank you.

[jimjam]

Thanks, great teaching. I really appreciate that you provide an index, it's really useful for an upcoming exam.

[jacoboribilik]

I love all these theorems on sequences they are so fun to learn and study.

[nyceric]

Camera man should stop playing with the camera, I feel like I'm on acid watching this video.

[jamesbra]

Wow are they teaching this in high school. I was still learning to tie my shoes in high school.

[chandankar]

I am preparing for my undergrad sem for real analysis-1 and it is quite helpful. Have you discussed limit points, bolzano weirstrass theorem for set, nested interval theorem in any of your videos ?
If any of these has been mentioned please reply me in which lecture number ?

[harshvardhansingh]

(a). At 50:59, professor can you please explain the alternative approach you are talking about ?
(b). Why in this question β ∉ S is necessary for the condition to hold ? β ∈ S still makes β the supremum, although β now may belong to set A (as it contains elements of S), still we can argue by the same approach that An < β implies there is an element between An and β and hence A is infinite?

[radnachtmadder]

Hi, my ultimate intent is to study probability and statistics in a rigorous fashion, id est through measure theory. I would like to know what sort of roadmap through real analysis and topology I am looking at and whether this is something I could study by myself.