Question: Consider a bipartite graph G with vertex classes A and B, where $|A|=|B|=n$. Show that if $|N(S)|>|S|$ for all $S \subset A$, then for any edge e of G, G contains a perfect matching which contains e.
Answer: So far I have shown that G contains a matching which contains e using Hall's Theorem however I am unsure about how to show that this matching is perfect. I have the following theorem:
Let G be a bipartite graph with vertex classes A and B and $|A|=|B|=n$. If $\delta (G) \geq n/2$, then G has a perfect matching.
But I don't know how to show the $\delta (G) \geq n/2$ part in this case?