<s>[INST] Floor Function. Let x and y be rational numbers let x and y be rational numbers. 
A. $\left \lfloor x \right \rfloor + \left \lfloor y \right \rfloor 
    = \left \lfloor x+y \right \rfloor$
B. $\left \lfloor x \right \rfloor + \left \lfloor y \right \rfloor 
    \le \left \lfloor x+y \right \rfloor$
C. $\left \lfloor x \right \rfloor + \left \lfloor y \right \rfloor 
    \ge \left \lfloor x+y \right \rfloor$
D. None of the above.
Can anyone please explain why the answer is B? Thank You!
 [/INST] If $x$ and $y$ are rational (or real) numbers, then there must exist numbers $h$ and $k$ such that $0 \le h,k < 1$,  $x = \left \lfloor x \right \rfloor + h$, and  $y = \left \lfloor y \right \rfloor + k$.
So:
\begin{align}
   \left \lfloor x + y \right \rfloor
   &= \left \lfloor
         \left \lfloor x \right \rfloor + h +
         \left \lfloor y \right \rfloor + k
     \right \rfloor \\
   &= \left \lfloor x \right \rfloor +
      \left \lfloor y \right \rfloor 
      + \left \lfloor h+k \right \rfloor\\
   &\ge \left \lfloor x \right \rfloor +
      \left \lfloor y \right \rfloor \\
\end{align}
</s>