Suprema and Infima of Combined Bounded Functions/Bounded Above

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

Let $c$ be a constant.

Let both $f$ and $g$ be bounded above on $S \subseteq \R$.

Then:
 * $\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$
 * $\ds \map {\sup_{x \mathop \in S} } {\map f x + \map g x} \le \map {\sup_{x \mathop \in S} } {\map f x} + \map {\sup_{x \mathop \in S} } {\map g x}$

where $\ds \map \sup {\map f x}$ is the supremum of $\map f x$.

Proof
First we show that:
 * $\ds \map {\sup_{x \mathop \in S} } {\map f x + c} = c + \map {\sup_{x \mathop \in S} } {\map f x}$

Let $T = \set {\map f x: x \in S}$.

Then:

Next we show that $\ds \map {\sup_{x \mathop \in S} } {\map f x + \map g x} \le \map {\sup_{x \mathop \in S} } {\map f x} + \map {\sup_{x \mathop \in S} } {\map g x}$:

Let:
 * $\ds H = \map {\sup_{x \mathop \in S} } {\map f x}$
 * $\ds K = \map {\sup_{x \mathop \in S} } {\map g x}$

Then:
 * $\forall x \in S: \map f x + \map g x \le H + K$

Hence $H + K$ is an upper bound for $\set {\map f x + \map g x: x \in S}$.

The result follows.