Equivalence of Definitions of Infimum of Real-Valued Function

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Theorem

Let $S \subseteq \R$ be a subset of the real numbers.

Let $f: S \to \R$ be a real function on $S$.

The following definitions of the concept of Infimum of Real-Valued Function are equivalent:

Definition 1

The infimum of $f$ on $S$ is defined by:

$\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left[{S}\right]$

where

$\inf f \left[{S}\right]$ is the infimum in $\R$ of the image of $S$ under $f$.

Definition 2

The infimum of $f$ on $S$ is defined as $\displaystyle \inf_{x \mathop \in S} f \left({x}\right) := k \in \R$ such that:

$(1): \quad \forall x \in S: k \le f \left({x}\right)$
$(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: f \left({x}\right) < k + \epsilon$


Proof

Definition 1 implies Definition 2

Let $k \in \R$ be an infimum of $f$ by definition 1.

Then from the definition:

$\text{(a)}: \quad k$ is a lower bound of $f \left({x}\right)$ in $\R$.
$\text{(b)}: \quad k \ge m$ for all lower bounds $m$ of $f \left({S}\right)$ in $\R$.

As $k$ is a lower bound it follows that:

$(1): \quad \forall x \in S: k \le f \left({x}\right)$


Now let $\epsilon \in \R_{>0}$.

Aiming for a contradiction, suppose $(2)$ were false:

$\forall x \in S: k + \epsilon \le f \left({x}\right)$


Then by definition, $k + \epsilon$ is a lower bound of $f$.

But by definition that means $k \ge k + \epsilon$.

So by Real Plus Epsilon:

$k > k$

From this contradiction we conclude that:

$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: f \left({x}\right) < k + \epsilon$

Thus $k$ is an infimum of $f$ by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $k$ be an infimum of $f$ by definition 2:

$(1): \quad \forall x \in S: k \le f \left({x}\right)$
$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: f \left({x}\right) < k + \epsilon$

From $(1)$ we have that $k$ is a lower bound of $f$.


Aiming for a contradiction, suppose that $k$ is not an infimum of $f$ by definition 1.

Then:

$\exists m \in \R, m > k: \forall x \in S: f \left({x}\right) \ge m$

Then:

$\exists \epsilon \in \R_{>0}: m = k + \epsilon$

Hence:

$\exists \epsilon \in \R_{>0}: \forall x \in S: k + \epsilon \le f \left({x}\right)$

This contradicts $(2)$.

Thus $k$ is an infimum of $f$ by definition 1.

$\blacksquare$