TITLE: If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?
QUESTION [7 upvotes]: Any help would be appreciated.
If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?

REPLY [2 votes]: I will show only the case where no vertex has an edge going to itself.  Assume a graph with $n$ vertices does not have at least two vertices of the same degree.  Then one vertex has degree one, second vertex has degree two,..., $nth$ vertex has degree $n$, for some ordering of the vertices.  Then the $nth$ vertex must be connected to $n$ other vertices.  But then there are $n+1$ vertices, so that we have a contradiction.