Consider proof 1 from König's lemma from wikipedia: https://proofwiki.org/wiki/König%27s_Tree_Lemma
At the end, they say they apply the axiom of dependent choice. I wonder on which set we apply the axiom: on the vertices of the $T$, or on our set $\{t_0,t_1,\dots\}$ which we have shown is indeed infinite. If it is the latter, then what relation did they use? If it's the former, same question really.. I'm confused because the specific relation of having inifnitely many descendants, and having a finite amount of ancestors which we can't take into acount anymore, doesn't seem to be true for all $t\in T$, so that's why I think it's the latter. But what's the relation? Is it that $t_iR t_j$ if $j=i+1$?