Sum of Two Well-Ordered Subsets is Well-Ordered – Set Theory

Question

Apologies if the answer is trivial, this is far from my domain.
In order to define the field of Hahn series, one needs the following fact: if $A,B$ are two well-ordered subsets of $\mathbb{R}$ (or any ordered group — with the induced order of course), the subset $A+B:=\{a+b\,|\,a\in A,b\in B\} $ is well-ordered. How does one see that?

Answer 1

Ramsey theory! Suppose $A + B$ is not well-ordered. Then there is a strictly decreasing sequence $a_1 + b_1 > a_2 + b_2 > \cdots$. Observe that for any $i < j$, either $a_i > a_j$ or $b_i > b_j$ (or both). Make a graph with vertex set $\mathbb{N}$ by putting an edge between $i$ and $j$ if $a_i > a_j$, for any $i < j$. By the countably infinite Ramsey theorem, there is either an infinite clique or an infinite anticlique, and hence either a strictly decreasing sequence in $A$ or a strictly decreasing sequence in $B$, contradiction.