 Infimum

For a nonempty set $A\subseteq \mathbb{R}$, a real number $\ell$ is its infimum precisely when it is a lower bound and every other lower bound is at most $\ell$. It is the greatest lower bound, not necessarily the minimum.

For $C=\{1/n:n\in {\mathbb{N}}_{>0}\}$, zero is a lower bound. Every positive candidate $\epsilon$ fails: the Archimedean property gives an integer $n>1/\epsilon$, hence $1/n<\epsilon$. Thus $\inf C=0$, although zero is never a member of $C$. This is the lower-edge counterpart of approximating an unattained supremum.