There's also a more elementary proof (although I also enjoy the use of Ramsey's countably infintie theorem!). First prove that an ordered set is well-ordered if and only if every sequence in it has a non-decreasing subsequence. Then given a sequence $(u_n)_{n \in \mathbb{N}}$ in $A+B$, for each $n \in \mathbb{N}$, pick the least $a_n \in A$ such that there exists a $b_n \in B$, which is then unique, with $u_n=a_n+b_n$. Extract a non-decreasing subsequence from $a$, then from $b \circ \varphi$ where $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ is the first "extraction map".

There's also a more elementary proof than the one given by Nik Weaver (although I also enjoy the use of the countably infinite Ramsey's theorem!). First prove that an ordered set is well-ordered if and only if every sequence in it has a non-decreasing subsequence. Then given a sequence $(u_n)_{n \in \mathbb{N}}$ in $A+B$, for each $n \in \mathbb{N}$, pick the least $a_n \in A$ such that there exists a $b_n \in B$, which is then unique, with $u_n=a_n+b_n$. Extract a non-decreasing subsequence from $a$, then from $b \circ \varphi$ where $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ is the first "extraction map".