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Real Order
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Optimization Problems
Real Limiting Bounds
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Greatest Lower Bounds


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.1

We call the unique greatest lower bound of a set (if it exists) the infimum.


We denote the infimum of a set $B \subset A$ by $\inf A$.

  1. Proof in future editions. ↩︎
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