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Are limit points of a sequence and an interval differently defined? I ask because I've been given the following definition of a limit point:

A limit point is one such that every of its neighborhoods contains infinitely many points of the set other than itself.

If that is so then this criteria applies only when we talk about an interval and it fails to give limit points of any sequence - for instance $(-1)^n$ has limit points at $1$ and $-1$, but these are not limit points according to the above definition.

What's the discrepancy between the limit points of a sequence and the definition given above?

Dheeraj Sonker
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    Well, tell us the definitions you've been given. I'd say that's pretty important. – Cameron Buie Jan 14 '16 at 17:49
  • The definition is every neighbourhood of the limit point should contain infinite many points of set other than itself. If that is so then this criteria applies only when we talk about an interval and it fails to give limit points of any sequence e.g. (-1)^n has limit points 1 & -1 but it does not confer to the above definition !! – Dheeraj Sonker Jan 14 '16 at 18:02
  • Link to similar or same question: https://math.stackexchange.com/questions/1071508/what-is-the-difference-between-the-limit-of-a-sequence-and-a-limit-point-of-a-se?rq=1 – Anders H Jan 08 '24 at 22:57

2 Answers2

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It's true that, taken naively, this definition does conflict with that of the limit points of a sequence. The problem is that one is counting the values $1$ and $-1$ only once, whereas they occur infinitely many times in the sequence. That is to say that we really should distinguish between the set $\{1,-1\}$ (which has no limit points) and the sequence $(-1)^n$ (which has two limit points).

A more analogous statement for sequences would be:

A limit point is a point such that the sequence has infinitely many terms in any neighborhood of it.

Which just basically repairs the statement for sequences by changing "distinct terms" to just "terms" - so that factors which are repeated infinitely often get included as limit points. So, all we need to do is be a bit careful about how we could when we have a set vs. how we count when we have a sequence.

Milo Brandt
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A limit point of a sequence $(a_n)_{n\to \infty}$ is defined as the point the sequence itself gets close. An interesting example of this is the sequence $(1)_{n\to \infty}$ approaches $1$.

While the limit point of a set is seen as a point in which every neighborhood of that point (the limit-point) contains points other than itself. More interesting the set $\{1\}$ has no limit point and the set, the set $\{1+\frac{1}{n}: n\in \mathbb{N}\}$ has $1$ as it's only limit point, and the set $\mathbb{R}$ has itself has it's set of limit points.

Conceptually they hold the same idea of closeness to a point. But mechnically they work a bit different. Sequences can only have one limit point (if we are talking about metric spaces), while Sets can have an infinite set limit points.

user160110
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    I believe "limit point (of a sequence)" usually refers to a point to which a subsequence converges - so one can have multiple limit points. (This is, of course, distinct from "limit of a sequence") – Milo Brandt Jan 14 '16 at 18:25
  • @MiloBrandt Oh, then the limit point of a sequence can have at most countably many points, which contrasts with the fact that the set of limit points of a set can have uncountably many points. The whole point of that, was to say, that when talking about limit points of sequences, you look for the end game [what does it approach]. Where as when you are talking about sets, you are strictly talking about what points(limit points) are infinitely close to that set (excluding the point itself). You can talk what happens to a sequence in the long run, not so much a set. – user160110 Jan 14 '16 at 18:34