An Introduction to Sequences in Real Analysis

[@idrankwhatt]

I have to say, I just passed my first Advanced Calc class and didn’t stumble onto your channel until now! I love the content you do and the fact that its such high production is a real plus. Keep it up man, you really make math digestible for everybody. (Btw In your ‘first year Ph.D. Program’ video I think you mentioned measure theory, as somebody interested in a statistics Ph.D., do you know if this class is common to take?)

[@JoelRosenfeld]

Congratulations on getting through analysis! If you have friends coming in behind you, send them my way! Measure theory is integral to statistics. Specifically, probability theory. You’ll definitely be learning a decent amount of measure theory in a statistics PhD. You’ll be concerned with finite measure spaces (because the maximum probability is 1). But most of the theory is the same.

[@DarinBrownSJDCMath]

4:43 I had to think about this for a while, but it seems that uniqueness of sequential limits lies strictly between the Hausdorff and T_1 separation axioms. Clearly, sequential limits are unique in every Hausdorff space. But the converse is not true -- take the cocountable topology on an uncountable set. Every two nonempty open sets meet, but also every convergent sequence is eventually constant. If a space is not T_1, then there exist distinct points p and q such that every open set including p must also include q. This implies that every sequence converging to q must also converge to p. Thus, every space in which sequential limits are unique must be T_1. The converse can be seen not to be true by considering the cofinite topology on an infinite set. Since every singleton is closed, the space is T_1, but any sequence of distinct elements converges to every point. It is true though, that being Hausdorff is equivalent to the uniqueness of limits of nets.

[@JoelRosenfeld]

Hmm… maybe I didn’t give this as much thought as I should have. Thanks for taking the time to write this up. It’s been a while since I really took a serious look at topology. I will trim out that statement when I get a chance.

[@AxiomTutor]

I like to often emphasize that definitions just state a property. Just because the property was stated doesn't tell you a priori how many things satisfy the definition -- it could be 1, 0, 2, or well ... you get the point. So the topology generated by a basis may not exist for some bases, for all you know, until you prove that it always exists.

[@okoyoso]

What is an example of a sequence converging to two different points when you're not Huasdorff?

[@JoelRosenfeld]

You can manufacture such a topology pretty easily. Basically, you can take the reals and for any open set that is not empty, union it with (0,1). This will give you a collection of sets that are closed under arbitrary unions and finite intersections, and you have the empty set and the set of all reals. Then the sequence 1/n converges to every element in (0,1).

[@gene-math]

good 👍thanks 😊

[@arwaabdelrahem2040]

can you add an arabic translation to your video

[@JoelRosenfeld]

I wish I could. But I don’t speak Arabic.