Problem 52E

Show that if x is a real number and n is an integer, then

Solution:

Step1

Given that

We have to show that if x is a real number and n is an integer, then

Step2

Suppose n<x ------------(1)

By using definition of Ceiling function “ it assigns to the real number y the smallest integer that is greater than or equal to y.”

Therefore,

-----------------(2)

From (1) and (2) we get

Hence, if nx then

----------------(3)

Step3

Suppose

By using definition of Ceiling Function

If then

This shows

If then

n< x -----------------(4)

From (3) and (4) we get

Step4

Suppose x<n ---------(5)

By using definition of floor function, “ The Floor Function assigns to the real number y the largest integer that is less than or equal to y.”

Therefore,

-------------------(6)

From (5) and (6) we get

If ----------------(7)

Step3

Suppose

Using Floor Function definition

If then

This shows that

If then --------(8)

From (7) and (8) we get