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.
Best Answer
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
$A=\{4,3,2,1,0\}$.
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 $A$.