Limit Points (Sequence and Neighborhood Definition) | Real Analysis

Your videos truly help me understand my lesson. Thank you.

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).

bro ur a gem.,!

best explanation that save my day☺🤟

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?

The concept has been well and clearly explained 😊

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.

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.

Excellent video! Can you do a proof by induction of the AM GM inequality? Would be really appreciated!

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.

An very important concept which is so well explained . Thank you sir for this video !

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! 😇