$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$

Solution 1:

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

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

Solution 2:

If $\left \lfloor x \right \rfloor = n$ then $x = n + r$ where $0 \leq r < 1$ and If $\left \lfloor y \right \rfloor = n'$ then $y = n' + r'$ where $0 \leq r' < 1$. Then $$x+y = n+n'+r+r'\geq n + n' = \left \lfloor x \right \rfloor + \left \lfloor y \right \rfloor $$.