 Matching covering maximum-degree vertices in a bipartite graph

A finite bipartite graph of positive maximum degree $\Delta$ has a matching in a graph covering every vertex of vertex degree $\Delta$. To cover the vertices of maximum degree in one side $A$, apply Hall's marriage theorem to that subset: for each subset $S$ of it, double counting gives $\Delta |S|\le \Delta |N(S)|$, where $N(S)$ is its graph neighbourhood. Construct the corresponding matching in a graph for the other side, then use the union of two matchings to cover both specified subsets simultaneously.