Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are bounded subsets of R. Prove each of the following. (a) sup{f(x)+g(x) : x € X}< sup{f(x) : x E X}+ sup{g(x): x € X }. (b) inf{f(x) :x € X}+inf{g(x): x € X} < inf{f(x)+g(x):xEX}. (c) If f(x) < g(x) for all x E X, then sup{f(x) : x € X} < sup{g(x):x € X}.

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