Application of Hall's marriage theorem

Question

Let $d\geq 1$ be an integer. Let $G$ be a bipartite graph with vertex sets $X,Y$ such that for all $S\subseteq X$, $|N(S) |\geq d|S|$. Show that we can match each element in $X$ to $d$ elements in $Y$.

Answer 1

Take a graph on vertex set $X'\cup Y$, where $X'$ consists of $d$ copies of $x$ for each $x\in X$. This graph satisfies Hall's condition (if we take a subset of $X'$ which meets copies of $k$ different vertices of $X$ then its neighbourhood has size at least $kd$, and the set itself has size at most $kd$). So it has a perfect matching, which corresponds to exactly what we want (match each $x\in X$ to the $d$ vertices matched to the vertices in $X'$ which correspond to $x$).