Let $X$ be a nonempty set, and let $f$ and $g$ be defined on $X$ and ave bounded ranges in $\mathbb{R}$. Show that:
$$\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x \in\ X\}$$
2026-04-02 19:00:56.1775156456
Show that $\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x \in\ X\}$
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$$\begin{gathered} f\left( x \right) + g\left( x \right) \leqslant \sup \left\{ {f\left( x \right):x \in X} \right\} + \sup \left\{ {g\left( x \right):x \in X} \right\},\,\,\forall x \in X \hfill \\ \Rightarrow \sup \left\{ {f\left( x \right) + g\left( x \right):x \in X} \right\} \leqslant \sup \left\{ {f\left( x \right):x \in X} \right\} + \sup \left\{ {g\left( x \right):x \in X} \right\} \hfill \\ \Rightarrow \inf \left\{ {f\left( x \right) + g\left( x \right):x \in X} \right\} \leqslant \sup \left\{ {f\left( x \right):x \in X} \right\} + \sup \left\{ {g\left( x \right):x \in X} \right\} \hfill \\ \end{gathered} $$