 Perfect matching criterion for a regular pair with minimum degree

Let $(X,Y)$ be a regular pair of vertex sets with parameter $ϵ$, equal class sizes $n$, density greater than $ϵ$, and every vertex having at least $ϵn$ neighbors across the pair. Then it has a perfect matching. To prove Hall's condition, take $S\subseteq X$. If $0<|S|\le ϵn$, one vertex already supplies at least $|S|$ neighbors. If $ϵn<|S|\le (1-ϵ)n$ and $|N(S)|<|S|$, the two large sets $S$ and $Y\setminus N(S)$ have density zero, contradicting regularity and the positive density surplus. Finally, if $|S|>(1-ϵ)n$, every vertex of $Y$ has a neighbor in $S$, so $N(S)=Y$. The empty subset satisfies Hall's condition directly.