I have to prove the following property of Floor function:
For any real number $x$, $x$ not being an integer, $\lfloor x \rfloor + \lfloor -x \rfloor = -1$.
Now, we know from the definition of floor that $\lfloor x \rfloor$ is the unique integer $n$ such that $n \leq x < n+1$. The trouble is writing $\lfloor -x \rfloor$. If I imagine a negative real number on the number line, it is obvious that $-n-1 \leq -x < -n$. Then, simply adding the two yields -1.
My problem is that I can't seem to arrive at $\lfloor -x \rfloor$ from the definition. For instance, if $x$ is a postive real number, then the floor is given by
$$n \leq x < n+1$$
Multiplying by -1 throughout,
$$-n \geq -x > -n-1$$
$$\Rightarrow -n-1 < x \leq -n$$
Feels like I'm almost there, but this does not match the definition (the $\leq$ has appeared on the upper bound). What am I doing wrong?