Suprema and Infima of Combined Bounded Functions

Theorem
Let $$f$$ and $$g$$ be real functions.

Let $$c$$ be a constant.

Bounded Above
Let both $$f$$ and $$g$$ be bounded above on $$S \subseteq \mathbb{R}$$.

Then:
 * $$\sup_{x \in S} \left({f \left({x}\right) + c}\right) = c + \sup_{x \in S} \left({f \left({x}\right)}\right)$$;
 * $$\sup_{x \in S} \left({f \left({x}\right) + g \left({x}\right)}\right) \le \sup_{x \in S} \left({f \left({x}\right)}\right) + \sup_{x \in S} \left({g \left({x}\right)}\right)$$

where $$\sup \left({f \left({x}\right)}\right)$$ is the supremum of $$f \left({x}\right)$$.

Bounded Below
Let both $$f$$ and $$g$$ be bounded below on $$S \subseteq \mathbb{R}$$.

Then:
 * $$\inf_{x \in S} \left({f \left({x}\right) + c}\right) = c + \inf_{x \in S} \left({f \left({x}\right)}\right)$$;
 * $$\inf_{x \in S} \left({f \left({x}\right) + g \left({x}\right)}\right) \ge \inf_{x \in S} \left({f \left({x}\right)}\right) + \inf_{x \in S} \left({g \left({x}\right)}\right)$$

where $$\inf \left({f \left({x}\right)}\right)$$ is the infimum of $$f \left({x}\right)$$.