Let $X$ be a set, let $\tau$ be a topology on $X$, let $\mathcal A \subset X$ be nonempty, and let $I:\mathcal A \to (-\infty,\infty]$ be an extended-real-valued functional. Define

\begin{align*} m:=\inf_{u\in\mathcal A} I[u]. \end{align*}

Assume that:

1. The infimum is finite:

\begin{align*} -\infty<m<\infty. \end{align*}

2. Every minimizing sequence in $\mathcal A$ has a $\tau$-convergent subsequence: if $(u_k)_{k=1}^{\infty}$ is a sequence in $\mathcal A$ such that $I[u_k]\to m$, then there exist a subsequence $(u_{k_j})_{j=1}^{\infty}$ and an element $u\in X$ such that $u_{k_j}\to u$ with respect to $\tau$. 3. The admissible class $\mathcal A$ is sequentially $\tau$-closed: whenever $(a_k)_{k=1}^{\infty}$ is a sequence in $\mathcal A$ and $a_k\to a$ with respect to $\tau$ for some $a\in X$, then $a\in\mathcal A$. 4. The functional $I$ is sequentially $\tau$-lower semicontinuous on $\mathcal A$: whenever $(a_k)_{k=1}^{\infty}$ is a sequence in $\mathcal A$, $a\in\mathcal A$, and $a_k\to a$ with respect to $\tau$, one has

\begin{align*} I[a]\le \liminf_{k\to\infty} I[a_k]. \end{align*}

Then there exists $u_*\in\mathcal A$ such that

\begin{align*} I[u_*]=\inf_{v\in\mathcal A} I[v]. \end{align*}