Closed Subset of Real Numbers with Lower Bound contains Infimum
Jump to navigation Jump to search
Let $A \subseteq \R$ such that $A$ is closed in $\R$ and $A \neq \varnothing$.
Let $A$ be bounded below.
Then $A$ contains its infimum.
- $\map \inf A \in \map \cl A$
- $A = \map \cl A$
Therefore $\map \inf A \in A$.