Let $X$ denote $\prod_{n=1}^\infty\mathbb{R}$, the Cartesian product of countably infinitely many copies of $\mathbb R$ (which is just the set of all infinite sequences of real numbers), endowed with the box topology. Now, let $X^+\subset X$ be the subset consisting of the sequences of strictly positive real numbers, and let $z$ denote the zero sequence, that is, the one whose terms are $z_i = 0$ for all $i$. Show that $z$ is in the closure of $X^+$, but there is no sequence of elements of $X^+$ converging to $z$.
I guess I did the first part. The closure of $X^+$ is $\bigcap_{\substack{\text{closed }S\,\subseteq X,\\ X^+\subseteq S}}S.$ But these subsets are of the form $\prod[-E,+\infty)$ for every $E\geq 0$, right? When $E = 0$, we have the required. Is it right? And what about the second part?