<s>[INST] 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 $x∈R$
 [/INST] Replacing $x$ by $x':=x+{1\over n}$ adds $1$ on the left hand side. On the right hand side the first term $\lfloor x\rfloor$ disappears, and on the far right the term $\lfloor x+1\rfloor$ appears. It follows that the right hand side increases by $1$ as well. 
Therefore it suffices to prove the formula for $0\leq x<{1\over n}$. Inspection immediately reveils that both sides are $=0$ in this case.
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