In this video lecture (Real analysis, HMC,2010, by Prof. Su) Professor Francis Su says (around 54:30) that "If a sequence $\{p_n\}$ converges to a point $p$ it does not necessarily mean $p$ is a limit point of the range of $\{p_n\}$." I'm not sure how that can hold (except in case of a constant sequence).

I'm not able to understand the difference between the set $\{p_n\}$ and the range of $\{p_n\}$ (which I understand is the set of all values attained by $p_n$). As per my understanding they're the same (except $\{p_n\}$ might contain some repeated values which the range of $p_n$ won't, e.g. the range of the sequence {1/2,1/2,1/2,1/2,1/3,1/3,1/3,1/3,1/4,1/4,...} would be {1/2,1/3,1/4,...}, or that of the constant sequence {1,1,1,...} would be {1}.

Based on this understanding, my reasoning is as follows: By the definition of a convergent sequence $\{p_n\}$ converging to $p$, for every $\epsilon \gt 0$ we can find an infinite number of terms of $\{p_n\}$ which lie at a distance less than $\epsilon $ from p, i.e. within an $\epsilon $-neighborhood of $p$. Hence every $\epsilon $-neighborhood of p contains an infinite number of points of the set $\{p_n\}$ other than itself ($p$). Hence $p$ is a limit point of $\{p_n\}$.

Now this would not hold only in case of a constant sequence, which converges, but any neighborhood of of its limit cannot contain any points in common with the sequence other than itself. Hence the limit won't be a limit point of the sequence.

Other than this special case, I cannot think of any situation where the limit of a convergent sequence is not also a limit point of the sequence.

Can anyone help? Thanks in advance.

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    $\begingroup$ Sequences which are eventually constant also work. $\endgroup$ – Joey Zou Aug 19 '16 at 6:57
  • $\begingroup$ No I absolutely agree that this is not true for a constant sequence, as I've explained. Just wanted to double check if there's any other case where this might hold, which I'm missing out. $\endgroup$ – Canine360 Aug 19 '16 at 6:57
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    $\begingroup$ You can also take any sequence that is eventually a constant sequence. $\endgroup$ – Cm7F7Bb Aug 19 '16 at 6:57
  • $\begingroup$ Range of $\{ p_n \}$? What is a range of a set? You can talk about a range of a map. I've never heard of the former before. $\endgroup$ – IAmNoOne Aug 19 '16 at 7:01
  • $\begingroup$ Thanks a ton JoeyZou and Cm7F7Bb. I didn't know the concept of an eventually constant sequence. Found out based on your guidance. Thanks! $\endgroup$ – Canine360 Aug 19 '16 at 7:11

Another [counter] example. Define a sequence in $(\mathcal{R},d)$ such that

$a_{n} := 5-n$ for $n \leq 5$ and

$a_{n} := 0$ for all $n > 5$

This sequence converges to 0 (zero). The range of this sequence viewed as a function from natural numbers to the real numbers is the following set


0 (zero) is not a limit point in this set since any $0 < r < 1$ will produce a neighborhood around 0 (zero) that will not include other points of this set $A$.


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