Let $X$ be a [topological space](/page/Topological%20Space), let $\mathcal A \subset X$ be nonempty, and let $I:\mathcal A \to \mathbb R\cup\{+\infty\}$ be bounded below in the extended-real sense. Suppose that every minimizing sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$, meaning every sequence satisfying $I[u_k]\to \inf_{v\in\mathcal A} I[v]$, has a subsequence $(u_{k_j})_{j=1}^{\infty}$ converging in $X$ to some point $u\in\mathcal A$. Suppose also that whenever $(u_k)_{k=1}^{\infty}$ is a minimizing sequence in $\mathcal A$ and $(u_{k_j})_{j=1}^{\infty}$ is such a convergent subsequence with limit $u\in\mathcal A$, one has $I[u]\le \liminf_{j\to\infty} I[u_{k_j}]$. Then there exists $u_*\in\mathcal A$ such that $I[u_*]=\inf_{v\in\mathcal A} I[v]$.