Consider a bipartite graph $G$ with vertex classes $A,B$ where $|B|=|A|=n$.
Use Hall's Theorem to show that if $|N(S)| \ge |S| + 1$ for all $S \subseteq A$ with $S \ne A$ and $S\ne\emptyset$, then for an edge $e$ of $G$, $G$ has a perfect matching which contains $e$.
So far I can only link this to the fact that Hall's theorem implies that a bipartite graph with vertex classes $|A|$ and $B$ of equal size has a perfect matching iff $|N(S)| \ge |S|$ for all $S \subseteq A$.
Otherwise I don't have any idea how to go about this.