<s>[INST] For all $x$ which are real numbers, prove that $\lfloor 2x\rfloor = \lfloor x\rfloor + \lfloor x+0.5\rfloor.$ 
For all $x$ which are real numbers, prove that
  $$\lfloor 2x\rfloor = \lfloor x\rfloor + \lfloor x+0.5\rfloor.$$

I know that
Let $\lfloor x\rfloor = n$
$n \leq x < n+1$
 [/INST] $n\leq x< n+1$ , then $f(x)=n$ for all $x$ 
I) If $n<x< \frac{2n+1}{2}$
$f(x)+f(x+0.5)= n+n=2n=f(2x)$
since $x+0.5<\frac{2n+1}{2} + \frac{1}{2}=n+1$
II) If  $n+1>x\geq \frac{2n+1}{2}$ 
$f(x)+f(x+0.5)= n+n+1=2n+1=f(2x)$
Since $x+0.5\geq\frac{2n+1}{2} + \frac{1}{2}=n+1$
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