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I've been stuck on this problem below and have a few more identical to it on my assignment. And yes I've Googled around and found a few links on here regarding to the density of $\mathbb{Q} \subseteq \mathbb{R}$, but I'm just not getting it.

Can somebody please dumb the steps down for me to prove it? It is an intro class, so I'm still trying to get down some of the vocab.

Prove that, if $y$ and $z$ are irrational numbers such that $y < z$, then there exists some $x \in \mathbb{Q}$ such that $y < x < z$.

(I'm unsure of the whole process, otherwise I'd attempt to show some work...)

Thank you

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  • $\begingroup$ I can write a proof for you. I can write a completely detailed proof. Or I can write a summary of a proof. But... there are literally hundreds, if not thousands, of books, articles, web pages, and videos that explain this, at every conceivable level of detail. So instead of asking for even more of this, why don't you ask a specific question about a specific step in a specific proof? Then we'd be happy to help you understand it. $\endgroup$
    – Zach Teitler
    Oct 6, 2017 at 1:06

1 Answer 1

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here's a sketch

Take the decimal expansion of each of them. They have to go on forever. There must be a point where they disagree.

Truncate the decimal expansion of the larger one at that point. You then have a rational which is between the two numbers.

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