Let $\alpha$ and $\beta$ be ordinals. For $f:\beta \rightarrow \alpha$, let $s(f)=\{\epsilon<\beta | f(\epsilon)\ne 0 \}$. Let $S(\beta,\alpha)=\{f|f:\beta \rightarrow \alpha $ and $s(f)$ is finite $\}$. Define $\prec$ on $S(\beta,\alpha)$ as follows: $f\prec g$ if and only if there is $\epsilon_0<\beta$ such that $f(\epsilon_0)<g(\epsilon_0)$ and $f(\epsilon)=g(\epsilon)$ for all $\epsilon>\epsilon_0$. Show that $(S(\beta,\alpha),\prec)$ is isomorphic to $(\alpha^{\beta},<)$.
Help please, I already provided the transfinite transitivity for $\alpha$ and $\beta$ but I still have problems with proof of well-order, any suggestions? I know it has to be related to the finite support of $ f $, but I can not find a way to conclude anything concrete.