(Solution) - Let X be a nonempty set and let f and -(2025 Original AI-Free Solution)

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Let X be a nonempty set, and let f and g be defined on X and have bounded ranges in R. Show that
Sup{f(x) + g(x) : x ? X} < supf{x) : x ? X} + sup{g(x) : x ? X}
and that
inf{f(x) : x ? X} + inf{g(x) : x ? X} < inf{f(x) + g(x) : x ? X}:
Give examples to show that each of these inequalities can be either equalities or strict
inequalities.