 Reverse-inclusion well-quasi-ordering of monomial ideals

Upward-closed subsets of ${\mathbb{N}}^r$, including the empty subset, form a well-quasi-ordering by reverse inclusion. For $r=1$, tails correspond to the well-order $\mathbb{N}\cup \{\infty \}$. Inductively slice in the last coordinate. The slices increase and eventually stabilize: their union has a finite minimal basis by the Dickson lemma, all of which has appeared in one finite slice. Encode an upset by the word of its pre-stabilization slices and its final slice. The Higman lemma and finite product closure of well-quasi-orderings compare these encodings in the reverse-inclusion slice order. A subsequence map $f$ has $f(k)\ge k$, so containment of the matched later slice implies containment of the slice at $k$; the final-slice comparison handles the tail. The entire earlier upset therefore contains the later upset.