If $A \subset \mathbb{R}^n$ is bounded an open then every sequence of $A$ has a subsequence that converges to a point of $A$

Question

Let $A \subset \mathbb{R}^n$ be bounded an open in $\mathbb{R}^n$. Is it true that every sequence of $A$ has a subsequence that converges to a point of $A$?

If $\{x_k\}$ is a sequence of points in $A$ then it is bounded because $A$ is bounded. By the Bolzano-Weierstrass theoren, if $\{x_k\}$ is bounded then it has a convergent subsequence $\{x_{k_r}\}$. However, does $\lim_{k \rightarrow \infty}x_{k_r}$ have to be in $A$?

Answer 1

No, this is false even for $n=1$.

Let $A$ be the open unit interval $]0,1[$, and let the series be $1/n$.