$\forall\,x,\,y\in\mathbb{R} : [x + y] = [x] + [y]$

Question

Can you please help me proving the identity

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$\forall\,x,\,y\in\mathbb{R} : [x + y] = [x] + [y]$, where $[\alpha]$ means the integer part of $\alpha$?

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I figured out that it holds for $x,\,y\in\mathbb{Z}$, $x = 0$ (or $y = 0$) or $x = y = 0$.

Any help is much appreciated.

Answer 1

This is false as stated. Take $x = 1.5$, $y = 1.5$. Then, $[x] = 1$, $[y] = 1$, but $$[x+y] = 3\ne 2 = [x] + [y]$$