How to prove this floor function equation? How can I prove the following equation?
$$
  \lfloor nx \rfloor =
  \lfloor x \rfloor + 
  \Big\lfloor x + \frac{1}{n} \Big\rfloor + 
  \Big\lfloor x + \frac{2}{n} \Big\rfloor + 
  \Big\lfloor x + \frac{3}{n} \Big\rfloor + 
  \Big\lfloor x + \frac{4}{n} \Big\rfloor + 
  \Big\lfloor x + \frac{5}{n} \Big\rfloor+ \dotsb + 
  \Big\lfloor x + \frac{n-1}{n} \Big\rfloor
$$
 $n∈N$ and $x∈R$

Try to rewrite $x$ as
$$x=\lfloor x \rfloor + \frac kn + \alpha$$
where $0\le k<n$ and $0\le\alpha<\frac1n$.
