Week 3 Exercise, due 12 October 2021

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Exercises 3.5 Week 3 Exercise, due 12 October 2021

1.

Prove that for every pair of real numbers \(x,y\in\RR\text{,}\) if \(x \lt y\text{,}\) then there exists a rational number \(q \in \QQ\) such that \(x \lt q \lt y\text{.}\)

Please write a careful proof, showing every step of your argument. I suggest using the Archimedean property of the real line to prove this!