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.