Understanding Limit Points: Definition & Meaning

[JG89]

In my textbook, it gives the following definition of a limit point:

"A point x is a limit point of a sequence x_1,x_2,... if every open interval containing x also contains points x_n for infinitely many n"

I am taking this to mean that every open interval containing x must also contain an infinite amount of other points from the sequence (though this doesn't necessarily mean that it contains every point of the sequence).

Is my understanding correct?

 

[CompuChip]

Yes. For example, 0 is a limit point of the sequence $a_n = 1/n$.
Although 0 is not itself in the sequence, any interval $(-\epsilon, \epsilon)$ contains infinitely many points from the sequence. For example, for $\epsilon = \frac{1}{\sqrt{10}}$, the interval does not contain a1, a2 and a3, but it does contain all other elements.

Note that the sequence needn't convert, if you take
$$a_{2n} = 1/n, a_{2n + 1} = 1$$
then 0 is still a limit point (and so is 1).

If I'm not mistaken, x is a limit point of a sequence S iff S has a subsequence S' which converges (Cauchy, and all) to S'. So basically, if part of the sequence has x as a limit. It's been a while since I did analysis though

 

[davyjones]

technically, limit pts have nothing to do with sequences
i personally think limit pt defines some special property of cercain set

 

[Tac-Tics]

technically, limit pts have nothing to do with sequences
i personally think limit pt defines some special property of cercain set

Limit points work just as well with sequences as with sets. Since the definition of a limit point does not refer at all to the order of the sequence, it's just a special case where the set is countably infinite.

 

[mathwonk]

your way of rephrasing the definition lost slightly some precision. The definition said that there are infiniutely many INDICES n for which x(n) belongs to the set, not infinitely many POINTS.

so the sequence could have repetitions. e.g. if the sequence is the constant sequence with all x(n) = x, then the definition is satisfied as you gave it for x to be a limit point.In an everyday space like the real numbers, and your language "interval" implies that is your space, it will be true that if every open interval about x contains an x(n) for even one n which is different from x, then also every interval contains an infinite number of them.

so you could say also that if every interval around x contains even one x(n) not equal to x, then x is a limit point. however if you want the constant sequence above to have x as a limit point, then you need the version you originally gave.