The original question is as follows:
For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$ and $\langle x \rangle = x - \lfloor x \rfloor$, that is, the fractional part of $x$. For arbitrary real numbers $x$, $y$ and $z$, only one of the following statements is correct. Which one is it?
- $\lfloor x + y + z \rfloor = \lfloor x \rfloor + \lfloor y \rfloor + \lfloor z \rfloor$
- $\lfloor x + y + z \rfloor = \lfloor x + y \rfloor + \lfloor z \rfloor = \lfloor x \rfloor + \lfloor y + z \rfloor = \lfloor x + z \rfloor + \lfloor y \rfloor$
- $\langle x +y + z \rangle = y + z - \lfloor y + z \rfloor + \langle x \rangle$
- $\lfloor x + y + z \rfloor = \lfloor x + y \rfloor + \lfloor z + \langle y + x \rangle \rfloor$
For options $(1)$, $(2)$ and $(3)$, I proved those relations to be false (actually special cases) from the definitions. Therefore, Option $(4)$ must be the correct one. In fact, it is.
However, my question is, how should I prove it as it were, if there were no other options? I tried using the definitions alone, but I got stuck. Should I consider proving it using cases? Or is there a simple approach?