I have the following question regarding Metric Spaces from Fred H. Croom's book Principles of Topology
Show that the limit of a convergent sequence of distinct points in a metric space is a limit point of the range of the sequence. Give an example to show that this is not true if the word "distinct" is omitted.
How would I approach this question properly? My attempt revolved around the following understanding.
If $(X,d)$ is a metric space and $\{x_n\}_{n=1}^{\infty}$ a sequence of points of $X$. Then $\{x_n\}_{n=1}^{\infty}$ converges to the point $x\in X$. Now if we assume $A\subset X$. A point $x \in X$ is a limit point of $A$ provided that there are infinitely many elements of the subset inside the neighborhood (open ball) $B_d(x,r)=\{a\in A: d(x,a)<\epsilon\}$. Thus, $x$ is a limit point of the sequence $\{x_n\}$ if given $\epsilon >0$, $x_n \approx x$ for infinitely many $n$.
Would I just have to simply show that both limit of the sequence $\{x_n\}$ and the limit point of $A$ are the same? Am I on the right track, and how would I properly state it?
What would then be an example to shows that this is not true if the word "distinct" is omiited?
I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide.
Best Answer
The problem is getting at the difference between distinct terms and distinct points. Consider a sequence $\langle x_n:n\in\Bbb N\rangle$; no matter what the sequence is, $x_3$ and $x_5$, for instance, are distinct terms of it. However, they may be the same point of $X$. Now let $x\in X$, and consider the sequence $\langle x_n:n\in\Bbb N\rangle$ defined by $x_n=x$ for each $n\in\Bbb N$.