Infimum of Singleton
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Then for all $a \in S$:
- $\inf \set a = a$
where $\inf$ denotes infimum.
Proof
Since $a \preceq a$, $a$ is a lower bound for $\set a$.
Let $b$ be another lower bound for $\set a$.
Then necessarily $b \preceq a$.
It follows that indeed:
- $\inf \set a = a$
as desired.
$\blacksquare$