This can be proven in a classical setting by considering the bar as an open covering of the compact topological space formula_25. Each sequence in the bar represents a basic open set of this space, and these basic open sets cover the space by assumption. By compactness, this cover has a finite subcover. The "N" of the fan theorem can be taken to be the length of the longest sequence whose basic open set is in the finite subcover. This topological proof can be used in classical mathematics to show that the following form of Kőnig's lemma holds: for any natural number "k", any infinite subtree of the tree formula_26 has an infinite path.