Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have is denoted by $\nu(H)$.
A set $M\subseteq E$ is said to be a matching if $M$ consists of pairwise disjoint members of $E$. Clearly, for any matching $M\subseteq E$ we have $|M|\leq \nu(H)$. We say that $H$ is balanced if there is a matching $M$ with $|M| = \nu(H)$.
Question. Let $H=(V,E)$ be a hypergraph such that for all finite $E_0\subseteq E$ the hypergraph $(V, E_0)$ is balanced. Does this imply that $H$ itself is balanced?