TITLE: Prove that in every graph with at least two vertices, there are at least two vertices which have the same degree?
QUESTION [7 upvotes]: I've tried deriving a proof for this using the handshaking lemma, but that was unsuccessful. Could someone please explain to me how they would do this proof?

REPLY [13 votes]: The possible degrees in a graph with $n$ vertices are $0,1,2, \dots, n-1$. Note that no graph with $n$ vertices can contain both a vertex of degree $0$ and a vertex of degree $n-1$, so in each case there are only $n-1$ possible degrees for $n$ vertices.