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Get stuck when approaching this problem. Thanks for any suggestions in advance.

Given two positive irrationals $a$ and $b$. Denote $f(x)=x-[\frac{x}{a}]a$ (brackets mean the floor function). Proof that for any given positive integer $x_0$, there exists another positive integer $x_1$ s.t. $f(x_0+b)>f(x_1+b)$.

Furthermore, given such $x_0$, is it possible to find out the minimum $x_1$ satisfying the above constraint?

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    HINT: write $f$ as a piecewise function defined in the intervals $na\le x <(n+1)a$. In addition you can change the variable to $y:=xa$. – 2017-02-25
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    In the case $a=b$, the constraint is reduced to $f(x_0)>f(x_1)$. However, I cannot prove it either. – 2017-02-25
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    I tried to break $f$ into pieces, but still failed to proof. – 2017-02-25

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