Let $\{p_n\}$ be a Cauchy sequence in $\mathbb{R}^k$. Let $K = cl \{p_n\}$ be an infinite compact subset of $\mathbb{R}^k$ (so $K$ is the closure of the set that consists of all distinct elements in the Cauchy sequence $\{p_n\}$). Let $p$ be a limit point of $K$.
- If we know that the neighborhood of p of radius 1 $N_1(p) \cap cl\{p_n\}$ is infinite, how can we prove that $N_1(p)\cap\{p_n\}$ is infinite?
- Is the set $\{p_n\}$ open or closed?
The proof of the first question is an intermediate step of the proof that Cauchy sequences are convergent in $\mathbb{R}^k$, but I don't know how to approach it.
For the second question, I think the set is open, cuase otherwise $\{p_n\} = cl\{p_n\}$ and the first question becomes trivial, but I'm not sure why it is open.
Let K be a countable subset of R$^n$.
Q1. If p is a limit point of K, U is an
open nhood of p, then U $\cap$ K is infinite.
If not, then an open nhood of p can be constructed that misses K except possibly at p.
So p is not a limit point of K.
If K is the set of points of Cauchy sequence s, then K is finite iff s is eventually constant.
Q2. K is not open because open subsets of
R$^k$ are uncountable. K is closed iff it
contains all of it limit points.