Open or Closed set in sup norm

Question

$ B=\big\{x= (x_n)_{n\in\mathbb{N}}:|x_n|<\epsilon\text{ for all }n\in \mathbb{N}\big\}$ , where $\epsilon>0$ is given in $(\ell^{\infty},||\cdot||_{\infty})$

How do I go about showing if this set is open or closed?

Answer 1

This set is neither open nor closed. To see that it is not open, consider the sequence $x_n=\epsilon\cdot (1-1/n)$, which is in $B$, and the sequences $y_n=x_n+1/m$ which satisfy $\|(x_n)-(y_n)\|_\infty=1/m\to 0$ as $m\to \infty$ but are not in $B$ as $|y_m|=\epsilon$. To see that it is not closed, consider the sequence $x_n=\epsilon-1/m$, which is in $B$ but as $m\to\infty$ this converges to the sequence $y_n=\epsilon$ which is not.