$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say that the largest values that could be added to $x$ and $y$ is $0.99$ and that doing so still made $\lfloor x\rfloor + \lfloor y\rfloor < \lfloor x+y\rfloor$. However, my answer was not accepted.

Let $x = \lfloor x \rfloor + \epsilon_1$ and $y = \lfloor y \rfloor + \epsilon_2$.
Then $\lfloor x + y \rfloor = \lfloor\lfloor x \rfloor + \epsilon_1 + \lfloor y \rfloor + \epsilon_2\rfloor = \lfloor x \rfloor + \lfloor y \rfloor + \lfloor \epsilon_1 + \epsilon_2 \rfloor \ge \lfloor x \rfloor + \lfloor y \rfloor$.
