Successive thinning proof of the infinite Ramsey theorem ($\mathbb{N}⟶(\mathbb{N}{)}_k^r$)

For finitely coloured $(r+1)$-element subsets, choose successive least points and use the inductive $r$-set theorem to thin the remaining reservoir after each choice. Every tuple with that least point then has one assigned colour. The infinite pigeonhole principle retains infinitely many least points with the same assigned colour, giving an infinite homogeneous set. The $r=1$ base case is the same pigeonhole principle. This proves the Ramsey theorem for r-sets for all finite $r$.