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?

• I know this page, actually I used this site to prove the formula in below mentioned way. – Matt Aug 16 '16 at 16:28
• This is not correct as written. For example, if $x = y = 1$, then $[x+y] = 2$ but $[y+x-[x]] = 1$. – Omnomnomnom Aug 16 '16 at 16:33
• You also have to add [x] = 1. Hence it becomes 2. – Matt Aug 16 '16 at 17:14
• Sorry I skipped it. I have edited the question. – Matt Aug 16 '16 at 17:15