I have the following question on topology:
Let $\mathbb R^\mathbb N = \prod_{n=0}^{\infty} \mathbb R$ with the box topology.
Show that for all $\bar x = \langle x_n \rangle_n$, the set $ A = \{\bar y :\langle x_n -y_n\rangle_n \to_{n \to \infty} 0 \} $ is open and closed.
I first tried to show that the set is open by showing that $A$ is a neighborhood of all $y\in A$, by showing that there exist open sets $V_i \subseteq \mathbb R$ such that
$$
\bar y \in \prod_{i=1}^{\infty}V_i \subseteq A
$$
but I'm not really sure if this is the right way. I would like some help or some guidance because this whole box/product topology doesn't sit right with me.
Thanks for the help!
Yes, finding these $V_i$ is the easiest way to show openness of $A$; it shows $y \in A$ is an interior point of $A$, for any such $y$.
Try $V_i = (y_i - \frac1i, y_i + \frac1i)$ e.g.
The closedness can also easily be shown by showing the openness of $A^\complement$ in a similar way.