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In my real analysis class, we have to determine whether or not the following is true. "There exists a sequence of real numbers that has infinite number of limit points." It then seemed to be true and an example is the sequence: 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,... I don't really understand how this sequence has infinite number of limit points. I guess the problem is that I don't get what a subsequence exactly means. I would appreciate it if somebody explains how we can have subsequences from this one that converge to infinite different numbers.

Sara
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    What is your definition for limit point? As I understand the definition, this would just be a discrete set and therefore not have any limit points. – Clayton Feb 16 '19 at 17:37
  • A number a is said to be a limit point of (xn) if there exists a subsequence of (xn) convergent to a. – Sara Feb 16 '19 at 17:41
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    As a matter of terminology, I was taught that a sequence could only have one limit point. What you are calling limit points, which are limit points of subsequences, I was taught to call accumulation points. I think your meaning is clear, but you may see the distinction made in the future. – Ross Millikan Feb 16 '19 at 17:45
  • I understand a sequence could only have one limit point when it is convergent. I mean the limit points of the subsequences. Thank you for your help. – Sara Feb 16 '19 at 18:00

2 Answers2

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Result: An element $a$ is a limit point of a sequence $a_n$ iff there exist a subsequence $a_{n_k}$ of $a_n$ converges to $a.$ Now we have

Subsequence $<1,1,\cdots>$ converges to $1$

Subsequence $<2,2,\cdots>$ converges to $2$

Subsequence $<n,n,\cdots>$ converges to $n$ and so on . So infinite limit points.

For yours second question of subsequence means. To obtain subsequence $a_{n_k}$ of a sequence $a_n$ just take sequence $n_k$ of suffixes to be in strictly increasing order. For example

sequence $<a_2, a_4, \cdots>$ for a subsequence as $<2,4,\cdots>$ is in strctly increasing order. But sequence $<a_2, a_6, a_4, a_{16} \cdots>$ is not a subsequence as $<2,6,4,16,\cdots>$ is not in strictly increasing order.

neelkanth
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A subsequence is just a collection of some of the terms of the sequence. Given your sequence $$1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,1,2,3,4,5,6,7\ldots$$ one of the subsequences is all the odd terms. It is $1,2,2,1,3,1,3,5,2,4,6\ldots$ but that is not interesting for this purpose. We are allowed to take any set of terms we want. If I make the subsequence consisting of all the $1$s, so the first, second, fourth,seventh, etc. terms, I get $1,1,1,1,1\ldots$, which has limit $1$. I can also make a subsequence consisting of all the $2$s, which has limit $2$, or a subsequence consisting of all the $k$s which has limit $k$. If you can convince yourself that any natural number appears an infinite number of times in the main sequence, then any natural number will be a limit point of the sequence and we have an infinite number of limit points.

Ross Millikan
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  • Thank you this really helps me a lot. I just want to know if there is a specific way we can describe these subsequences. I mean if we take only even terms, we write n1=2, n2=4, n3=6... Is it just enough to write these subsequences like 1,1,1,1,1,... – Sara Feb 16 '19 at 17:52
  • Just saying you take all the $1$s is a sufficient description, but we can describe it explicitly. Note that the first appearance of a number $k$ is $T_k=\frac 12k(k+1)$ the $k^{th}$ triangular number. The other appearances of $k$ are at positions $T_n+k$ with $n \gt k$ – Ross Millikan Feb 16 '19 at 19:14