Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.16

16. Show that -[-x] is the least integer greater than or equal to x when x is a real number.
Solution:  We define the greatest integer, denoted by [x] , as the largest integer less than or equal to x satisfying the inequation
[x]<=x<[x]+1 …..(1)
Where x is a real number
We assume that
y=-x
Then, from (1) we get,
[y]<=y<[y]+1
We take negative sign which will reverse the inequality, so
-[y]>=-y>-[y]-1
Therefore, we get,
-[-x]>=-(-x)>-[-x]-1
-[-x]>=x>-([-x]+1)
This implies -[-x] that is the least integer greater than or equal to x whenever x is a real number.

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