Let $\mathbb{N}^*$ be the set of all finite words over the natural numbers, that is $\mathbb{N}^*=\{n_0n_1\ldots n_{i-1}\mid i\in\mathbb{N},\ n_j\in\mathbb{N}\text{ for all }j<i\}$. On this set, we can impose a lexicographic ordering $\preceq$. Note that we can view $\mathbb N^*$ as an infinitely branching tree $T_\mathbb{N}$, with the empty word at the root, all "single-letter" words at the second level, and so on.
After giving it some thought, it seems clear that a subset $A\subseteq\mathbb N^*$ is finite iff all subsets of $A$ have a minimal and a maximal element with respect to $\preceq$.
In the case of the set of words over a two-letter alphabet, say $\{0,1\}^*$, which we can view as the full binary tree in a way similar to $T_\mathbb{N}$, we can use König's Lemma to prove the above statement. However, how would we go about proving this statement if the graph by which we can view our set is no longer finitely branching?
EDIT. The proof over $\{0,1\}^*$ goes something like: suppose $A$ is an infinite subset of $\{0,1\}^*$. Then by König's Lemma, we can conclude that there must hence be an infinite path in the full binary tree with infinitely many elements of $A$ on it, and thus, taking $B\subseteq A$ to be the set of all points on this path, we see that $B$ has no maximal element with respect to $\preceq$, and hence by contradiction we see the statement is true.
The direction that a finite subset has a maximal element is easy. There is some maximum length word and some maximum integer used. Your subset is then a subset of a finitely branching tree. We also need to prove that any infinite $A$ will have a subset without a maximal element. If there are an infinite number of numbers in the first position, all of $A$ has no maximal element. If there are finite number of numbers in the first position, you can apply König's Lemma because the first layer is finitely branching. There will be some first position number that starts an infinite tree. Take the subset consisting of all members of $A$ with that number in the first position. You either reach a position with infinitely many branches or you find an infinite branch.