Theorem (König's lemma)   Let $T$ be a rooted directed tree. If each vertex has finite degree but there are arbitrarily long rooted paths in $T$ then $T$ contains an infinite path.

Proof. For each $n\ge 1$ let $P_n$ be a rooted path in $T$ of length $n$ and let $c_n$ be the child of the root appearing in $P_n$ By assumption, the set $\{c_n\mid n\ge 1\}$ is finite. Since the set $\{P_n\mid n\ge 1\}$ is infinite, the pigeonhole principle implies that there is a child $c$ of the root such that $c=c_n$ for infinitely many $n$

Now let us look at the subtree $T_c$ of $T$ rooted at $c$ Each vertex has finite degree, and since there are paths $P_n$ of arbitrarily long length in $T$ passing through $c$ there are arbitrarily long paths in $T_c$ rooted at $c$ Hence if $T$ satisfies the hypothesis of the lemma, the root has a child $c$ such that $T_c$ also satisfies the hypothesis of the lemma. Hence we may inductively build up a path in $T$ of infinite length, at each stage selecting a child so that the subtree rooted at that vertex still has arbitrarily long paths.