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

Let ##\alpha \in \mathbb{R}## and ##n \in \mathbb{N}##. Show that exists a number ##m \in \mathbb{Z}## such that ##\alpha - \frac {m}{n} \leq \frac{1}{2n}## (1).

3. The attempt at a solution

If I take ##\alpha= [\alpha] +(\alpha)## with ##[\alpha]=m## (=the integer part) and ##(\alpha)=\frac{1}{2n}##(=the fractional part) I must have an equality in (1). Substituting, I obtain ##m(n-1)=0## and for ##n=1## I can always find a solution to the problem. Is this demonstration correct? How can I find a more general one?

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# Homework Help: Simple demonstration with real, rational and integers

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