<s>[INST] How to prove property of greatest integer function: 
How can we prove that
  $$\lfloor x + y \rfloor = \lfloor y + x -  \lfloor x \rfloor \rfloor + \lfloor x \rfloor$$
  for all real $x$, where $ \lfloor x \rfloor$ denotes greatest integer less than or equal to $x$?

I was able to prove that $\lfloor x + y \rfloor =  \lfloor x \rfloor + \lfloor y \rfloor $ or $\lfloor x + y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor + 1$ by using the property that $\lfloor x \rfloor = m$ means $m \le x \lt m + 1$ but cannot prove above one by any means. I have tried a lot, can any one please help and please give a simple proof?
 [/INST] If $n$ is an integer and $t$ any real number, then it is straightforward to show that$\lfloor t + n \rfloor = \lfloor t \rfloor + n.$
Therefore, since $\lfloor x \rfloor$ is an integer,
\begin{align*}
\quad &\lfloor y + x -  \lfloor x \rfloor \rfloor + \lfloor x \rfloor \\
= &\lfloor y + x \rfloor - \lfloor x \rfloor + \lfloor x \rfloor \\
= &\lfloor x + y \rfloor.
\end{align*}
</s>