- #1

cmajor47

- 57

- 0

## Homework Statement

Prove that in any interval there exists a rational z.

## Homework Equations

## The Attempt at a Solution

My professor wrote this for me when trying to explain how to prove this:

[itex]a \notin Q[/itex], [itex]\epsilon[/itex] rational

[itex][r, s]\in a[/itex]

[itex]l([r, s])<\frac{\epsilon}{2}[/itex]

[itex]s-a<\frac{\epsilon}{2}[/itex]

[itex]s-\frac{\epsilon}{2}<a[/itex]

[itex]s-\frac{\epsilon}{2}+\epsilon<a+\epsilon[/itex]

[itex]s<s+\epsilon<a+\epsilon[/itex]

I don't see why you use [itex][r, s]\in a[/itex] or how my professor went from [itex]s-\frac{\epsilon}{2}+\epsilon<a+\epsilon[/itex] to [itex]s<s+\epsilon<a+\epsilon[/itex]

I also don't know where to go from here to show what needs to be proved.

I'd really appreciate any help.

Last edited: