Let $G = (V, E)$ be a bipartite graph with $n$ nodes on each side. Show that if the degree of each vertex is greater than $n/2$, then $G$ has a perfect matching.
My thoughts are to use the Marriage Theorem which my book states as
A bipartite graph has a perfect matching if and only if $|A| = |B|$ and for any subset of (say) $k$ nodes of $A$ there are at least $k$ nodes of $B$ that are connected to at least one of them.
It is easy to show that any subset of $A$ of fewer than $n/2$ nodes is connected to at least $n/2$ nodes in $B$. However, I do not know how to conclude anything about bigger subsets of $A$.