From the given we know that these exist: .
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This means that is an upper bound of .
Can you finish?
(Let R represent the real numbers) Let D be a nonempty set and suppose f: D --> R and g: D-->R. Define f+g: D-->R by (f+g)(x) = f(x)+g(x). If f(D) and g(D) are bounded above then PROVE sup[(f+g)(D)] is less than or equal to sup f(D) + sup g(D).