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Obviously, for any $\epsilon >0$, there exist $m,n\in \mathbb{N}$ such that$$|\sqrt{2}-\frac{n}{m}|<\epsilon \; \textrm{.}$$ Is it also true that for all $\epsilon >0$, there exist $m,n\in \mathbb{N}$ such that$$|\sqrt{2}m-n|<\epsilon \; \textrm{?}$$ If so, does it also hold for transcendental numbers?

Mathmo123
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Philmore
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1 Answers1

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Here

https://en.wikipedia.org/wiki/Diophantine_approximation

it is shown that every irrational number $\alpha$ satisfies

$$|\alpha-\frac{p}{q}|<\frac{1}{q^2}$$

for infinite many pairs $(p,q)$. If you multiply with $q$, you see that the answer to your question is "yes".

Peter
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