Theorem (Proving the infimum)

To show that an element $a\in A$ is the infimum or that $a= \inf A$ then we need need to show:

1. Approximation property of the infimum: $\forall\epsilon>0,\exists a_{0} \in A \mbox{ s.t. } a_{0}<\inf A+\epsilon$
2. Lower bound property: $\forall a\in A, a\ge \inf A$