19. In this problem, we will show that the ceiling and floor functions are well-defined. For a real number x, the ceiling of x, denoted x "rounds up to the nearest integer. For example,= 4,1/2= 1, and 0= 0. The floor of x, denoted x rounds down to the nearest integer .For this problem, we must assume The Archimedean Property for the real numbers. That is, we assume that for every real number x, there exists a natural number n such that n > x. Your will learn more about this when you take a real analysis class.

(a) Use the Well-Ordering Property to prove that for every xR with x >0, there exists a unique smallest yZ such that y x.

(b) Use the Archimedean Property, together with part (a) to prove that for everyxRwithx0, there exists a unique yZ such that yx. [Hint: Apply the Archimedean Property tox.](c) Taking parts (a) and (b) together, we may define the ceiling function :RZ to be the function which assigns to a real number x the unique smallest integer y such that yx. Now define the floor function in terms of the ceiling function. That is, write a formula for x (which we have not yet defined) using [-] (which we have defined).