Prove that if x is a real number then
[2x] = [x] + [x + ½ ]
Ans: Let us consider x be any real number. It comprises two parts: integer and fraction. With no loss of any type of generality, fraction part can all time be made +ve. For instance, -1.3 can be written as -2 + 0.7.
Here now write x = a + b, and [x] = a (integer part only of the real x). The fraction part b requires to considered in two cases:
0 < b < 0.5 and 0.5 ≤ b < 1.
Case 1: 0 < b < 0.5; In this case [2x] = 2a, and [x] +[x + .5] = a + a = 2a
Case 2: 0.5 ≤ b < 1; In this case [2x] = 2a + 1, and [x] +[x + .5] = a + (a + 1) = 2a +1
Hence [2x] = [x] + [x + .5]