Proof about floor function: $\lfloor x\rfloor+\lfloor x + \frac{1}{n} \rfloor+\cdots + \lfloor x + \frac{(n-1)}{n} \rfloor = \lfloor nx \rfloor$
Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$