(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that in any interval there exists an irrational z.

2. Relevant equations

3. 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.

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# Homework Help: Prove in any interval there exists an irrational z

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