Introduction to Higher Mathematics - Lecture 15: Sequences and Functions

We finish up our discussion of real numbers with a look at sequences, bringing the real numbers to completion.

[goldjoinery]

I love this series of lectures. I've been wanting to find a good source of advanced mathematics for a long time and the quality of your content is great! I'm a 13-year-old student from Singapore, and I find my curriculum to be going at too slow a pace. Your videos are so well-scripted and clear, that I can often put the concepts you go over to near-immediate use. Thank you for making these lectures, Mr. Shillito.

[MrMiaumee]

Minor typo in the 3rd bullet point at 10:14. Should be "a_m < a_n for all m > n". Impressive graphic work and intro to analysis though.

[IslandForestPlains]

I like your "ultimum" - hope it dissipates across the boundaries of this course

[jbragg]

Thank you so much for this video, I got halfway through the Baby Rudin as they call it, and a lot of concepts eluded me like limit points, cauchy sequences etc... I understand them now. Thanks a lot :)

[JAMESLEEROC]

10:14 If a sequence is decreasing iff, I think for the 3rd point, it should be "am is less than an, for all m greater than n".

[TheKivifreak]

A sequence is an ordered list of elements from a multiset. Sequences can have an element multiple times. Great series btw, really enjoying it!

[BillShillito]

Point taken. Must have misspoken, my apologies! It turns out the terminology we use for sequences and the terminology we use for functions are often the same, so luckily, the concepts are parallel. :)

[BillShillito]

Fixed. Switched one too many inequality signs. :P Thank you!

[drewm]

The term "extremum" means "any point at which the the value of a function is the largest or smallest" (source: 1989 Encyclopedia Britannica) so that looks like the same thing as the ultimum.

[janmican]

Why don't you use 'limit' instead of supremum and infimum? It fits the same definition... Thank you for the great series Bill, your presentations are great!

[ningwang]

I love both ultimum and whenevever[27:06].

[SeekerofTao]

Im a Math Major starting real analysis next semester, and I am going to use Ultimum. Gonna make it a thing.

[andypetsch]

08:40 You say "A function is increasing..." which is somewhat confusing. As you mentioned some minutes before, one may see a sequence as a function. But since we are only talking about sequences in this part of the video it would have been clearer if you would have used "A sequence..." instead of "A function...".

[marjanminou]

Bill Shillito, creator of the Ultimum. If I ever see someone claiming it's their idea, they will suffer! :D

[ryanmike]

What about a sequence in which each term is the square root of its indexed natural number (y=sqrt(x)). I think this would be a Cauchy sequence because the terms are getting closer to each other. However, isn't it the case that this sequence in fact does not converge? Wouldn't this contradict the fact that every Cauchy sequence of real numbers is convergent. I'm sure that I've over looked something pretty simple so please help me find what it is!

Also, thanks so much for making these videos!

[LaureanoLuna]

I'd say the first inequality symbol in that definition should be 'greater than or equal to' instead of the simply 'greater than' on the screen.

[paulderrickbingan]

Sir please do examples as you did with groups