Infimum in Ordered Subset

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Theorem

Let $L = \left({S, \preceq}\right)$ be an ordered set.

Let $R = \left({T, \preceq'}\right)$ be an ordered subset of $L$.

Let $X \subseteq T$ such that

$X$ admits an infimum in $L$.


Then $\inf_L X \in T$ if and only if

$X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$


Proof

By definition of ordered subset:

$T \subseteq S$

and

$\forall x, y \in T: x \preceq' y \iff x \preceq y$

Sufficient Condition

Let $\inf_L X \in T$.

By definition of infimum:

$\inf_L X$ is lower bound for $X$ in $L$.

By definition of $\preceq'$:

$\inf_L X$ is lower bound for $X$ in $R$.

We will prove that

$\forall x \in T: x$ is lower bound for $X$ in $R \implies x \preceq' \inf_L X$

Let $x \in T$ such that

$x$ is lower bound for $X$ in $R$.

By definition of $\preceq'$:

$x$ is lower bound for $X$ in $L$.

By definition of infimum:

$x \preceq \inf_L X$

Thus by definition of $\preceq'$:

$x \preceq' \inf_L X$

$\Box$

Hence $X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$

$\Box$

Necessary Condition

assume that

$X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$

By definition of infimum:

$\inf_R X \in T$

Thus by assumption:

$\inf_L X \in T$

$\blacksquare$


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