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Lower Bound

Last updated Nov 1, 2022

# Definition

Let $(P, \leq)$ be a Partial Ordering

and let

A \subset P

.

x \in P

is an Upper Bound for

A

if

\forall a \in A

,

p \geq a

.

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Compact Sets in the Order Topology contain their Extrema

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.... Then $\sup\limits K$ and $\inf\limits K$ both exist. Furthermore, $\sup\limits K \in K$ and $\inf\limits K \in K$ . Proof Recall that . Therefore, $K$ has an and a . Because the and , we know $K$ is......

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Infimum

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Tightly Bounded Set

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