Let x and y be real numbers. Prove that the greatest integer function satisfies the follow...

Let x and y be real numbers. Prove that the greatest integer function satisfies the following properties:

(a) [x + n] = [x] + n for any integer n.

(b) [x] + [−x] = 0 or − 1, according as x is an integer or not.

[Hint: Write x = [x] + θ, with 0 ≤ θ<1, so that −x = − [x] − 1 + (1 − θ).]

(c) [x] + [y] ≤ [x + y] and, when x and y are positive, [x][y] ≤ [xy].

(d) [x/n] = [[x]/n] for any positive integer n.

[Hint: Let x/n = [x/n]+ θ, where 0 ≤ θ < 1; then [x] = n[x/n] + [nθ].]

(e) [nm/k] > n[m/k] for positive integers, n, m, k.

(f) [x] + [y] + [x + y] ≤ [2x] + [2y].

[Hint: Let x = [x] + θ, 0 ≤ θ < 1, and y = [y] + θ′, 0 < θ′ < 1. Consider cases in which neither, one, or both of θ and θ′ are greater than or equal to .]