Greatest Lower Bounds
Definition
Suppose $(A, \leq)$ is a partially ordered set.
A lower bound for $B
\subset A$ is an element $a \in A$ satisfying
\[
a \leq b \quad \text{for all } b \in B
\]
In words, $a$ is a predecessor of every
element of $B$.
A set is bounded from
below if it has a lower bound.
A greatest lower bound
for $B$ is an element $c \in A$ so that $c$
is a lower bound and $c < a$ for all other
lower bounds $a$.
If there is a greatest lower bound it is
unique.
We call the unique greatest lower bound of a
set (if it exists) the
infimum.
Notation
We denote the infimum of a set $B \subset A$
by $\inf A$.