Convergent sequence in box topology Given $P = \left\{a\in R^R: a_n > 0 \ \forall  n\in R\right\}$. Let 0 $= (0_n)_{n\in R}$. Assume $R^R$ has the box topology. Prove that:
(a) 0 is in Cl($P$) (i.e, closure of $P$)
(b) No sequence in $P$ converges to $0$.
My Progress: For part (a), using a classic result stating that between every two real numbers there exists another real numbers, we see that for any $\epsilon > 0$, there must exist a point $a$ with all positive coordinates such that every coordinates of $a$ is contained in some neighborhood of coordinates of point $0$, each of the form $(0,\epsilon)$. Therefore, crossing all of those neighborhood for all coordinates of point $a$, we infer that the point $a$ must be inside the intersection of the point $0$  with $P$ for any $\epsilon > 0$. By a well-known theorem, 0 is in Cl($P$).
For part $(b)$, I haven't gotten anywhere yet:P Hope someone can help me with this part, and please verify if my proof for part (a) is correct.
 A: I’m going to assume that you made no typo and that the space really is $\Bbb R^{\Bbb R}$ with the box topology; only minor changes are needed if you meant to write $\Bbb R^{\Bbb N}$.
Your proof of (a) isn’t quite right, because the $\epsilon$ can be different on each coordinate. A basic open nbhd of $\mathbf{0}$ in the box topology on $\Bbb R^{\Bbb R}$ looks like
$$\prod_{r\in\Bbb R}(-\epsilon_r,\epsilon_r)=\left\{x\in\Bbb R^{\Bbb R}:|x_r|<\epsilon_r\text{ for each }r\in\Bbb R\right\}\;,\tag{1}$$
where $\epsilon_r>0$ for each $r\in\Bbb R$. Your basic idea is sound, though: for each $r\in\Bbb R$ we can choose an $a_r\in(0,\epsilon_r)$, and the point $a=\langle a_r:r\in\Bbb R\rangle$ is in $P\cap\prod_{r\in\Bbb R}(-\epsilon_r,\epsilon_r)$.
HINT for (b): Suppose that $\langle a^{(n)}:n\in\Bbb N\rangle$ is a sequence in $P$, where $a^{(n)}=\langle a_r^{(n)}:r\in\Bbb R\rangle$. For $n\in\Bbb N$ let $\epsilon_n=\frac12a_n^{(n)}$, and for $r\in\Bbb R\setminus\Bbb N$ let $\epsilon_r=1$. Now consider the open nbhd of $\mathbf{0}$ in $(1)$ using these values for the numbers $\epsilon_r$.
