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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