Number of sequences defined on a set

Question

Given a set $D\subset \mathbb{R}$ and $c\in D$ how many sequences $f_r:\mathbb{N} \rightarrow D$ can be defined such that $\lim\limits_{n\rightarrow \infty} f(n)=c$

Answer 1

HINT: If $D=\{c\}$, there is only one. If $|D|>1$, and $c$ is an isolated point of $D$, there are countably infinitely many. If $c$ is a limit point of $D$, there are $\mathfrak{c}=|\Bbb R|=|\wp(\Bbb N)|$. To prove these three results you will probably want to use the following facts.

• If $c$ is isolated, any sequence in $D$ converging to $c$ must be eventually constant.
• If $\sigma=\langle x_n:n\in\Bbb N\rangle$ is a sequence of distinct points converging to $c$, so is every subsequence of $\sigma$.