How to prove that $\lfloor a\rfloor+\lfloor b\rfloor\leq \lfloor a+b\rfloor$ We have the floor function $F(x)=[x]$ such that $F(2.5)=[2.5]=2, F(-3.5)=[-3.5]=-4, F(2)=[2]=2$. 
How can I prove the following property of floor function:
$$[a]+[b] \le [a+b]$$

By the definition of $[x]$ as the greatest integer less than or equal to $x$, we have
$$[a]\le a\quad\text{and}\quad [b]\le b$$
and thus
$$[a]+[b]\le a+b$$
Now $[a]+[b]$ is certainly an integer (because it's the sum of two integers).  Therefore, by the definition of $[x]$ as the greatest integer less than or equal to $x$, we have
$$[a]+[b]\le [a+b]$$
