A short constructive proof of the Erdős–Gallai characterization of graphic lists

Erdős and Gallai proved that a nonincreasing list ( d 1 , … , d n ) of nonnegative integers is the list of degrees of a graph (with no loops or multi-edges) if and only if the sum is even and the list satisfies ∑ i = 1 k d i ≤ k ( k − 1 ) + ∑ i = k + 1 n min { k , d i } for 1 ≤ k ≤ n . We give a short constructive proof of the characterization.