So the problem is the following
"Does the equality $\lfloor$nx$\rfloor$=n$\lfloor$x$\rfloor$ hold for integers n$\le$2"
I have been trying to look at the Hermite's identity to see if that could help but I couldn't get anywhere with that and I'm confused at to what approach I could use to simplify the problem. I know that you can for example use a=$\lfloor$x$\rfloor$ and use the doubleinequality a$\le$x$\lt$a+1 to argue different cases. Could someone help to shed some light at to what of the basics I'm missing to solve this problem?
We can rewrite $x=\lfloor x \rfloor + y$ where $ 0 \leq y < 1$.
The equation becomes $$\lfloor n \lfloor x \rfloor + ny \rfloor = n \lfloor x \rfloor$$
But since the first term is an integer, we can pull it out as such:
$$ n \lfloor x \rfloor + \lfloor ny \rfloor = n \lfloor x \rfloor$$
So we see that the equality holds iff $\lfloor ny \rfloor = 0$. However, this is not always the case - consider $y = \frac{1}{2}, n=2$.