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The solution of a problem I had to work on states that :

$$\mid y\Big[\frac{x}{y}\Big]-a\mid <\mid x-a\mid + \mid y \mid \quad x,y\in R$$

Where $[x]$ is the integer part of $x$. I don't know to get this inequality... I only have : $$\mid y\Big[\frac{x}{y}\Big]-a\mid <\mid x + a\mid$$ Thanks for your help !

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Write $\left[\frac{x}{y}\right]= \frac{x}{y} - \left\{\frac{x}{y}\right\}$, where $\left\{\frac{x}{y}\right\} < 1$ is the fractional part. Then apply the triangle inequality.

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    $\begingroup$ Oh yes I did not think about the fractional part, it works thanks ! $\endgroup$ – Dicordi Jun 3 at 20:04

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