Could someone please check my work

Let D be a nonempty set and suppose that f : D -> R and g : D - R. Define the function f + 9: D - R by (f + g)(x) = f(x)+g(x). 1. If f(D) and g( D) are bounded above, then prove that (f + 9)(D) is bounded above. Suppose D is a nonempty set and suppose that f : D -> R and g: D - R, and f(D) and g(D) are bounded above. Since D is nonempty, then f(D) , g(D) , and (f + 9)(D) are nonempty. By the completeness axiom, since f(D) and g(D) are nonempty and bounded above, then they must exist and be real numbers. f(x)