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m For i small (e.g. i=1) the fraction does not have to be strictly bigger than zero. Thus changed less than to less than or eq. Tag: Reverted |
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Since <math>f</math> is periodic with period <math>1</math> and <math>0 \in \Q,</math> it suffices to check all irrational points in <math>I=(0,1).\;</math> Assume now <math>\varepsilon > 0,\; i \in \N</math> and <math>x_0 \in I \setminus \Q.</math> According to the [[Archimedean property]] of the reals, there exists <math>r \in \N</math> with <math>1/r < \varepsilon ,</math> and there exist <math>\; k_i \in \N,</math> such that
for <math>i = 1, \ldots, r</math> we have <math>0
The minimal distance of <math>x_0</math> to its ''i''-th lower and upper bounds equals
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