Floor Function

Floor and ceiling functions

Say we have $\alpha$, then $\lfloor\alpha\rfloor$ is the greatest integer less than or equal to $\alpha$ and $\lceil\alpha\rceil$ is the smallest integer greater than or equal to $\alpha$. Finally, $\{ \alpha \}$, the fractional part of $\alpha$, is the part of the real number $\alpha$ after the decimal.

Floor Function

Floor function of $x$ where $x$ is a real number, denoted by $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. For example, $\lfloor 3.1415\rfloor=3$, $\lfloor-2\rfloor=-2$ and $\lfloor5\rfloor=5$.

If you got the two questions correctly, congratulations!

Prove that for real $x$ and integer $n$, we have $\lfloor x+n\rfloor=\lfloor x\rfloor+n$.

Prove that for real $x,y$, we have $\lfloor x+y\rfloor\geq\lfloor x\rfloor+\lfloor y\rfloor$.