Suprema and Infima of Combined Bounded Functions/Bounded Above

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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:

\(\ds \map {\sup_{x \mathop \in S} } {\map f x + c}\) \(=\) \(\ds \map {\sup_{y \mathop \in T} } {y + c}\)
\(\ds \) \(=\) \(\ds c + \map {\sup_{y \mathop \in T} } y\) Supremum Plus Constant
\(\ds \) \(=\) \(\ds c + \map {\sup_{x \mathop \in S} } {\map f x}\)


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.

$\blacksquare$


Warning

The equalities do not apply in general.

Let us take as an example:

$S = \closedint {-1} 1$
$\map f x = x$
$\map g x = -x$

where $f$ and $g$ are real functions defined on $\R$.

Then:

$\ds \map {\sup_{x \mathop \in S} } {\map f x} = \map {\sup_{x \mathop \in S} } {\map g x} = 1$
$\ds \map {\inf_{x \mathop \in S} } {\map f x} = \map {\inf_{x \mathop \in S} } {\map g x} = -1$

So:

$\ds \map {\sup_{x \mathop \in S} } {\map f x} + \map {\sup_{x \mathop \in S} } {\map g x} = 2$
$\ds \map {\inf_{x \mathop \in S} } {\map f x} + \map {\inf_{x \mathop \in S} } {\map g x} = -2$

However:

$\forall x \in S: \map f x + \map g x = x + \paren {-x} = 0$

So:

$\ds \map {\sup_{x \mathop \in S} } {\map f x + \map g x} = \map {\inf_{x \mathop \in S} } {\map f x + \map g x} = 0$

and it is immediately clear that the equality does not hold.


Sources

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