Let f x dxe be the ceiling function Prove that if x R and m

Let f (x) = dxe be the ceiling function.

Prove that if x R and m Z, then dx+me = dxe+m. Hint: use the property that dxe = n if and only if n1 < x n

Solution

Ceiling function is the function which gives the value equal to the next integer of that

Example = ceiling (1.1) = 2 and ceiling (1.9) = 2

dxe = n when n1 < x n

thus dx + me = will be eqaul to cieling function of x which lies in above range + m

which is equal to the dxe +m because here also we are adding m just to the cieling of that function.