<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] Let $ x = i + d $ where $ 0 \le d < 1 $
$ \lfloor 2x \rfloor = \lfloor 2i + 2d \rfloor = 2i + \lfloor 2d \rfloor $
$ \lfloor x \rfloor + \lfloor x + 0.5 \rfloor = 2i + \lfloor d \rfloor + \lfloor d + 0.5 \rfloor $
So we reduced the problem to just show for $ d $
Suppose $ 0 \le d < 0.5 $, then both sides are just $ 2d $
Suppose $ 0.5 \le d < 1 $, then both sides are just $ d + 1 $.
QED $ \blacksquare $
</s>