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I was studying on sequences and their limit and limit points.

I got stuck in visualising this following problem, which seemed so simple.

The problem is: Suppose I have a sequence of a discrete values such that after a particular value the sequence is eventually constant. The range set is finite.

So the limit of the sequence is that particular constant value because if I take any open interval around that value, it consists of infinitely many terms of that sequence. This is fine.

But I am confused to see how is that the limit point of the range set? How do we say that by the definition of the limit point?

MB17
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2 Answers2

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A sequence $x_n : \mathbb{N} \rightarrow \mathbb{R}$ converges to $x$ if for all $\varepsilon>0$, there exists and $N_\varepsilon$ so that $n > N_\varepsilon$ implies $|x_n - x| \le \varepsilon$. There are many equivalent definitions, but this is the standard one. The idea is that if you go out far enough into the sequence, all the remaining terms are eventually clustered as tightly as you want around some particular number, $x$. So if the sequence is eventually constant, finding $N_\varepsilon$ is very easy.

On the other hand, there are lots of other definitions. There is something called $\lim \sup$, which is $$ \sup_{N>0} \lim_{n \ge N} x_n $$ and $\lim \inf$, which is $$ \inf_{N>0} \lim_{n \ge N} x_n $$ that bound the range of the sequences as we throw away any number of initial terms that you want: the sequence is eventually below $\lim \sup$ and above $\lim \inf$. This gets closer to the idea of what is happening ``in the range space''. But if a sequence $x_n \rightarrow x$, the $\lim \sup$ and $\lim \inf$ are equal, and eventually the range is as small a ball around $x$ as you want. That's the whole idea of convergence.

  • I get this. But what about the limit point of the range set? As per it's definition, one must have a non empty set on the intersection with any open interval around that constant and the Range set-{constant}. But as far as I see, it gives me an empty intersection. But I know every limit is a limit point What am I missing? – MB17 Jun 09 '20 at 14:52
  • Define "limit point of the range set" in mathematical notation, and tell me how it is distinct from the definition of a limit –  Jun 09 '20 at 14:57
  • That is what I am stuck at . If I have set of natural numbers, a set with discrete values, there is no limit point because if I take any open interval around an element belonging to N, it gives me empty intersection ( as per the definition of limit point) But how to check the limit point for a range set? And for the "limit", I guess we have have for any open interval around that element, has to have infinitely many terms of that sequence. – MB17 Jun 09 '20 at 16:04
  • If we take any open interval around that constant element we have infinitely many terms of that sequence, thus we can say that that constant is the "limit". But when we check for the limit point of that range set, taking any open interval around that constant and it's intersection gives us empty, indicating that constant is not the limit point. But however it must be. Why? – MB17 Jun 09 '20 at 16:10
  • If $x_n \rightarrow x$ and $\varepsilon >0$, there are a countably infinite number of terms of the sequence in the open ball $(x-\varepsilon,x+\varepsilon)$ for any $\varepsilon >0$. But there might not even be one element of the sequence that takes that value, like $1/n \rightarrow 0$. Why do you want to take an intersection of the limit and the image of the tail of the sequence? It is almost like you are trying to think about sequences like a topologist in terms of open sets but using real analysis definitions based on sequences. –  Jun 09 '20 at 17:09
  • Here's an alternative definition of convergence: $x_n \rightarrow x$ iff every neighborhood of $x$ contains all but finitely many of the points $x_n$. So take any open ball around $x$, if I can tighten the ball and throw out finitely terms of the sequence, what I have cannot be the limit. If I can keep tightening and tightening until there's a unique element left, that's the limit. So if you have a non-convergent sequences with multiple convergent subsequences, the problem is that they have different limits and tightening the ball throws out an infinite number of terms. But if all subsequences –  Jun 09 '20 at 17:16
  • have the same limit, then that nbhd of $x$ can be tightened down as much as you like towards a single point. –  Jun 09 '20 at 17:17
  • Let us continue this discussion in chat. – MB17 Jun 09 '20 at 17:36
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Note: A similar answered question is here: What is the difference between the limit of a sequence and a limit point of a set?

For the sake of documentation, I will try to answer this question in a different way, borrowing the terminology of the previous answer.

You describe a sequence $x_n$ that converges to $x$, i.e. $x_n\to x$ or equivalently $\lim_{(n\to\infty)}x_n=x$. Furthermore, in your example there exists an $N$ such that for all $n>N$, $x_n=x$. As you say, this is fine and within the definition of a convergent sequence.

Not being all familiar with this term, I take it that the range set of sequence $(x_n)$ is the set of every member of the sequence; $R=\bigcup \{x_n\}=\{x_n$ for all $n \}$.

Now, (in my understanding), limit points of sets are defined with intervals in mind, as being inner or limit points of intervals. The neighborhood requirement in the definition leads to such points being accepted while discrete/singular points are not accepted.

In this example, $R$ is finite and discrete. $R$ contains no intervals of any size $>0$. Therefore it contains no limit points such as they are defined for sets.

So it is all in good order that your sequence has a limit (which is its only limit point) but its range set has no limit point. These are two different concepts of "limit point".

Update: for more information: https://en.m.wikipedia.org/wiki/Accumulation_point. The lead section of this article [currently] includes a summary of the distinction between "limit points" of sets vs sequences.

Anders H
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  • How does one link questions? Similar question with answers here: 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 23:01
  • How to... link... OK, apparently by adding links to the Q or an A. Doing it in a comment does not properly make the questions linked? – Anders H Jan 08 '24 at 23:16