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## 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 [tex]\in[/tex] E. Denote the set

Na(E) = {x > 0|(a-x, a+x) [tex]\subset[/tex]E}

(a) Explain why Na(E) is nonempty.

(b) Prove that if x [tex]\in[/tex] Na(E) then [a-x, a+x] [tex]\subset[/tex] E.

(c) Prove that if [a-x; a+x] [tex]\subset[/tex] E, then there is a y [tex]\in[/tex] Na(E) satisfying y > x.

(d) Show that Na(E) is not bounded above (argue by contradiction).

(e) Prove that E = R.

## Homework Equations

## 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[tex]\in[/tex]R so that (a-x)<y<(a+x).

b)

I was thinking, since x is in Na(E) that would mean that x[tex]\in[/tex] E and since a and x are both in E, by definition of a set, that would mean that

a-x [tex]\in[/tex] E, and

a+x [tex]\in[/tex] 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] [tex]\subset[/tex] E

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