ビーティ数列
m = ⌊ ( ⌊ m r ⌋ + 1 ) r ⌋ ⟺ m < ( ⌊ m r ⌋ + 1 ) r < m + 1 ⟺ m r < ⌊ m r ⌋ + 1 < m + 1 r ⟺ ⌊ m r ⌋ + 1 − 1 r < m r < ⌊ m r ⌋ + 1 ⟺ 1 − 1 r < m r − ⌊ m r ⌋ = [ m r ] 1 . {\displaystyle {\begin{aligned}m=\left\lfloor \left(\left\lfloor {\frac {m}{r}}\right\rfloor +1\right)r\right\rfloor &\iff m<\left(\left\lfloor {\frac {m}{r}}\right\rfloor +1\right)r<m+1\\&\iff {\frac {m}{r}}<\left\lfloor {\frac {m}{r}}\right\rfloor +1<{\frac {m+1}{r}}\iff \left\lfloor {\frac {m}{r}}\right\rfloor +1-{\frac {1}{r}}<{\frac {m}{r}}<\left\lfloor {\frac {m}{r}}\right\rfloor +1\\&\iff 1-{\frac {1}{r}}<{\frac {m}{r}}-\left\lfloor {\frac {m}{r}}\right\rfloor =\left[{\frac {m}{r}}\right]_{1}.\end{aligned}}}