Prove that 1 Lxj 1x for any real number xSolutionCase 1 x

Prove that: 1 - Lxj = [1-x] for any real number x

Solution

Case 1. x is an integer

1-floor(x)=1-x=ceiling(1-x)

BEcause , 1-x is also an intger

Case 2: x is not an integer

Let, x=k+e , e is fractional part of x

0<e<1 , k is an integer,k>=0

1- floor(x)=1-k

1-x=1-k-e=-k+1-e

0<e<1

-1<-e<0

0<1-e<1

Hence, 1-e is fractional part of 1-x and -k is integer part

Hence, floor(1-x)=-k and ceiling(1-x)=-k+1

Hence, ceiling(1-x)=1-k=1- floor(x)

Hence proved.