Labelled version of the almost bounded case of S. B. Rao's degree sequence conjecture

S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been proved by Chudnovsky and Seymour. We give an independent short proof of the labelled version of the almost bounded case of S. B. Rao’s conjecture, the case where we have a bound on the degree, but allow a bounded number of vertices to have unbounded degree.