 Approximation property of an infimum

Let $A\subseteq \mathbb{R}$ be nonempty and bounded below. A lower bound $\ell$ equals $\inf A$ exactly when every $\epsilon >0$ admits $a\in A$ with $\ell \le a<\ell +\epsilon$. Otherwise $\ell +\epsilon$ would be a larger lower bound. Taking tolerances $1/n$ gives a sequence of members converging to $\ell$. For a real-valued objective with finite infimum, this supplies a minimizing sequence without assuming that a global minimizer exists.