Supremum of Function is less than Supremum of Greater Function

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

Let $S$ be a subset of $\Dom f \cap \Dom g$.

Let $\map f x \le \map g x$ for every $x \in S$.

Let $\ds \sup_{x \mathop \in S} \map g x$ exist.

Then $\ds \sup_{x \mathop \in S} \map f x$ exists and:


 * $\ds \sup_{x \mathop \in S} \map f x \le \sup_{x \mathop \in S} \map g x$.

Proof
We have:

Supremum of Sum equals Sum of Suprema also gives that $\sup f$ and $\sup \paren {g - f}$ exist.

We have: