# Open and Closed sets

## Homework Statement

Let E be a nonempty subset of R, and assume that E is both open
and closed. Since E is nonempty there is an element a $$\in$$ E. De note the set
Na(E) = {x > 0|(a-x, a+x) $$\subset$$E}

(a) Explain why Na(E) is nonempty.
(b) Prove that if x $$\in$$ Na(E) then [a-x, a+x] $$\subset$$ E.
(c) Prove that if [a-x; a+x] $$\subset$$ E, then there is a y $$\in$$ Na(E) satisfying y > x.
(d) Show that Na(E) is not bounded above (argue by contradiction).
(e) Prove that E = R.

## The Attempt at a Solution

a) I don't know if I can just say for (a-x)<(a+x) there has to be an y$$\in$$R so that (a-x)<y<(a+x).

b)
I was thinking, since x is in Na(E) that would mean that x$$\in$$ E and since a and x are both in E, by definition of a set, that would mean that
a-x $$\in$$ E, and
a+x $$\in$$ E
and since (a-x, a+x) is contained in E and now (a-x) and (a+x) are in E, that would mean:
[a-x,a+x] $$\subset$$ E

The rest I'm not too sure about, any help would be great!