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.
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