Lexicographic order on $\alpha^\beta$ is well-ordered Suppose that $\alpha$ and $\beta$ are ordinals, and define the following order on $\alpha^\beta$, the set of all functions $\beta\to\alpha$ with finite support: $$f\,R\,g\iff\exists x\in\beta\,(f(x)<g(x)\land\forall y\in\beta\,(x<y\to f(y)=g(y)))$$ I wish to prove that $R$ is a well order of $\alpha^\beta$. I have already shown that $R$ is a total order on $\alpha^\beta$, so it remains to prove that if $S$ is a nonempty subset of $\alpha^\beta$ then it contains an $R$-minimal element, but I'm at a loss as for how to select this element and the Wikipedia section on this is rather terse with respect to actual proofs.

You can do it by induction on $\beta$. Assume that it's true for $\alpha^\gamma$ with $\gamma<\beta$. If the zero function is in $S$, we're done. Otherwise, for $f\in S$ let $\eta(f)=\max\operatorname{supp}(f)$, and let $\eta_0=\min\\{\eta(f):f\in S\\}$. Let $\xi_0=\min\\{f(\eta_0):f\in S\text{ and }\eta(f)=\eta_0\\}$, and let $$S_0=\\{f\upharpoonright\eta_0:f\in S\text{ and }\eta(f)=\eta_0\text{ and }f(\eta_0)=\xi_0\\}\;.$$

Now let $f\in S$ be such that $\eta(f)=\eta_0$, $f(\eta_0)=\xi_0$, and $f\upharpoonright\eta_0=\min S_0$, and verify that $f=\min S$.