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## Homework Statement

Prove the set [itex] A= \{ \frac{p^2}{q^2}: p<q , p,q \in \mathbb{N} \} [/itex]

is dense on the interval [0,1]

## The Attempt at a Solution

ok so if I have 2 arbitrary reals a and b on the interval [0,1] and a<b

I could easily pick p such that [itex] \frac{1}{p^2}<b-a [/itex]

and I can do this by the Archimedean principle. But now I need to pick a natural number

that puts me in between a and b. And this natural number need to have a natural number

when square rooted and be less than p^2 . I guess im not so worried about being able to pick

a number whose square puts me in between a and b . because I could just pick p to be larger which would allow more options for q . but im not sure how to make q<p.