Let D be a nonempty set and suppose that f : D → R and g : D → R. Define the function f + g : D → R by ( f + g)(x) = f (x) + g (x). (a) If f (D) and g (D) are bounded above, then prove that ( f +...

Let D be a nonempty set and suppose that f : D → R and g : D → R. Define the function f + g : D → R by ( f + g)(x) = f (x) + g (x).

(a) If f (D) and g (D) are bounded above, then prove that ( f + g)(D) is bounded above and sup [( f + g)(D)] ≤ sup f (D) + sup g (D).

(b) Find an example to show that a strict inequality in part (a) may occur.

(c) State and prove the analog of part (a) for infima.