Rao's degree sequence conjecture

Let us say two (simple) graphs G,G′ are degree-equivalent if they have the same vertex set, and for every vertex, its degrees in G and in G′ are equal. In the early 1980's, S.B. Rao made the conjecture that in any infinite set of graphs, there exist two of them, say G and H, such that H is isomorphic to an induced subgraph of some graph that is degree-equivalent to G. We prove this conjecture.