TITLE: Why do graph degree sequences always have at least one number repeated?
QUESTION [17 upvotes]: Why do graph degree sequences always have at least one number repeated?  
$(1, 2, 2, 3)$ = Valid, as you can see, because the $2$ is repeated.
$(1, 2, 3)$ = Not possible to construct a graph with because all numbers are unique.  
I know that the graph degree sum must be even, but in this case, both are even.
Our professor said it had to do with Combinatorics, but we never touched that topic.
EDIT: Loops are not allowed.

REPLY [2 votes]: For graphs which aren't simple it's false generally. Example:
$ V = \{1,2,3\}$, $ E = \{\{1,2\}, \{2,3\}, \{2,3\}\} $  , where $V$ is the set of vertices and $E$ is the multiset of edges. We have $d(1)=1, d(2)=3, d(3)=2$.