<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] If you take a closer look, you will notice that the second term within each floored term will be less than one.
E.g in $$\left\lfloor x+\frac{4}{n}\right\rfloor$$As you can see $\frac{4}{n}$ is less than one.
So, we can conclude that each term will be reduced to $\lfloor x\rfloor$ if $n$ and $x$ are integers.
We will get
$$\lfloor nx\rfloor = n\lfloor x\rfloor$$
Since $n$ and $x$ are integers, we can remove the floor function.
Hence proved.
This identity will not be true for all values of $n$ and $x$ if they can be any number other than integers.  Try putting any fractional value.
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