To what point or points does the sequence converge?

Question

In the finite complement topology on $\mathbb{R}$ , to what point or points does the sequence $\{x_{n}\} $ converge ? Here is my solution-

Let $G$ be a neighbourhood of $x$ in $\mathbb{R}$. i.e., $X-G$ is finite . Then sequence $\{x_{n}\} \in G$ for all but finitely many $n$. Therefore $\{x_{n}\}\to x$.

What to do next actaully I am confused

Look at this answer: For a finite complement topology, to which point or points does the sequence converge — Karthik Kannan, May 25 '20 at 14:37

score 1 · Accepted Answer · answered May 25 '20 at 16:27

1

Let $G$ be a neighbourhood of $x$ in $\mathbb{R}$. i.e., $X-G$ is finite . Then sequence $\{x_{n}\} \in G$ for all but finitely many $n$. Therefore $\{x_{n}\}\to x$. Since every point of $\mathbb{R}$ is distinct. There are only a finite number of points of sequence that may not be lie in $G$. So the sequence converge to every pont of $\mathbb{R}_{FC}$

answered May 25 '20 at 16:27

What about the sequence $(0,1,0,1,0,\ldots)$? It does not converge to anything. – JasonJones Feb 16 '21 at 22:32

To what point or points does the sequence converge?

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