Limit Points (Sequence and Neighborhood Definition) | Real Analysis
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- Опубликовано: 18 сен 2024
- Limit points, accumulation points, cluster points, whatever you call them - that's today's subject. We'll define limit points in two ways. First we'll discuss the sequence definition of a limit point of a set. Then we'll discuss the neighborhood definition of a limit point of a set. We will see examples of limit points and non-examples, and conclude with a proof that the two definition of limit point are equivalent. #realanalysis
Limit points can be used to define what are called closed sets also, check it out! • All About Closed Sets ...
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Your videos truly help me understand my lesson. Thank you.
Glad I could help!
Extremely helpful, thanks!
Great topic! So important.
Now, can you tie this discussion to point set topology and open vs closed sets and then to compact sets and the associated theorems for the set of real numbers and then for metric spaces and finally for general topological spaces? That would be a master class on this super important topic and require several videos. Can't wait to see it! 😇
Thanks Ezra! I'm working my way through the real analysis playlist, so no general topological spaces yet! I'd like to create full playlists for every undergraduate course before moving on to grad topics, but topology is probably near the bottom of the line in that list - partly because I never took a course on it so I need to study it more. But I never took a course on graph theory either, so it's nothing more than a minor inconvenience. But indeed the proof connecting closed sets and limit points is next! Then the Heine-Borel theorem whenever I have time to sit down and record it!
@@WrathofMath Awesome! 👍
My experience with Real Analysis is limited to what is covered in Calculus (single-varible through vector) to validate the basic ideas of differentiation and integration. Since Calculus is so geometric (visual), what makes Real Analysis so difficult?
So, the limit point dont have be a point in the set?, in your example (0,2),0 is a limit point which is not in (0,2).
Exactly!
An very important concept which is so well explained . Thank you sir for this video !
Thank you!
bro ur a gem.,!
Thank you! I'm doing my best!
Thanks for the video. Question -- 4:16 why is 1 a limit point of the set (0, 2)?
I had interpreted your explanation at 1:20 as "the set cannot contain the limit point", though to your credit, after rewatching that portion I realize you actually said "the *sequence* cannot contain the limit point."
But if that's the case, and if 1 can be a limit point of a set like (0, 2), can't literally every number in that set be its limit point? Given the notion of the set being "everywhere dense" (such that, for *any point a* in (0, 2) , there is an infinitely dense subset in the 𝛿 neighborhood of a)? E.g., why can't 1.5 also be a limit point, since there's an infinite sequence 1.49, 1.499, 1.499 ... approaching 1.5?
Perhaps I'm confusing concepts and terminology somewhere here. If so, please let me know. Thanks.
It’s from my understanding of what he said that every number in the set IS a limit point. So you’re right, 1.5 is a limit point of (0,2). See the video at 4:55.
1 is a limit point but not a boundary. Boundaries are usually like that
Thanks
Glad to help, thanks for watching!
Excellent video! Can you do a proof by induction of the AM GM inequality? Would be really appreciated!
Thank you! I'd be happy to, but it might be a little while until I have time!
The concept has been well and clearly explained 😊
Thank you for watching!
Thanks Wrath of Math, I have seen definitions of limit points defined as “if any neighborhood of x contains a point in S\{x} then x is a limit point of S. I know it is very similar to the definition you provided but it seems much weaker because it only requires 1 point (in S )different than x in every neighborhood of x for x to qualify as a limit point. Could you please clarify this in another video?
Thanks for watching and the question! The definition you describe appears the same as what I used, you'll have to rephrase it if there is a difference I am missing.
The definition used in the video is: A point x is a limit point of S if every neighborhood of x intersects A at some point other than x. In other words, every neighborhood of x must contain at least one point that is also in A.
The definition you describe, as I understand it, is: A point x is a limit point of S if every neighborhood of x contains a point (other than x) that is also in S.
You used the word "any" at the beginning of your definition, which would mean "at least one", but I assume you meant every.
Nice presentation.
Thank you!
I wish I could double-like this video!
I wish I could double like this comment!
Thank you for this great video on this topic. I have a question for you.
There is a convergent sequence defined on the set X. A finite number (for example, 10) of elements of that sequence forms a closed set, which also has an open neighborhood. Is it possible that the limit of this sequence lies outside that closed set and its open neighborhood? Or should it necessarily reside in this closed set?
I would be very happy if you answer my question.
Thanks for watching! I'm not sure I understand your question, the open neighborhood part in particular. Where is this open neighborhood? It sounds like you're saying the closed set has it, but you also said the closed set consists of a finite number of elements, which can't ever be open (aside from the empty set).
@@WrathofMath let me ask my question more clearly:
There is a space X and there exists a sequence in this space. In this space X, there exists a closed set composed of a certain finite number of elements of the sequence, and it also has an open neighborhood. This sequence has a unique limit, but it belongs neither to this closed set nor to its open neighborhood. Can this limit point be separated by a function from that closed set?
Well-explained and visuallised video. Do you know a proof of the fact that if the limit of the sequence a_n+1/ a_n is between 0 and 1 then the sequence a_n converges to 0.
if all an>0 must be in condition thats ratio test
i'm not sure what "intersects" means but i'm guessing "intersects" means something like this: A intersects B := cardinality of A∩B is bigger than 0
why is your def so much different than taos doesn't even seem like that's the def at all
Thank you for this great video on this topic. I have a question for you.
There is a convergent sequence defined on the set X. A finite number (for example, 10) of elements of that sequence forms a closed set, which also has an open neighborhood. Is it possible that the limit of this sequence lies outside that closed set and its open neighborhood? Or should it necessarily reside in this closed set?
I would be very happy if you answer my question.