Show-that-x-nx-n-where-n-N-x-R-

Question Number 119048 by Hassen_Timol last updated on 21/Oct/20

Show that :

⌊ x ⌋ = ⌊ \frac{⌊ n x ⌋}{n} ⌋ where n \in N^{*} x \in R

Answered by mr W last updated on 21/Oct/20

s a y x = m + f w i t h 0 ⩽ f < 1

⌊ x ⌋ = m

⌊ n x ⌋ = n m + ⌊ n f ⌋

0 ⩽ n f < n

0 ⩽ ⌊ n f ⌋ < n

0 ⩽ \frac{⌊ n f ⌋}{n} < 1

⌊ \frac{⌊ n f ⌋}{n} ⌋ = 0

\frac{⌊ n x ⌋}{n} = m + \frac{⌊ n f ⌋}{n}

⌊ \frac{⌊ n x ⌋}{n} ⌋ = m + ⌊ \frac{⌊ n f ⌋}{n} ⌋ = m = ⌊ x ⌋

\Rightarrow ⌊ x ⌋ = ⌊ \frac{⌊ n x ⌋}{n} ⌋ p r o v e d

Commented by Hassen_Timol last updated on 21/Oct/20

Thank you so much Sir, may god bless you