I was demonstrating the following Lemma:
König's Infinity Lemma: Let $ T $ be a tree all of whose levels are finite and nonempty. Then there is a path through T.
I asked myself: Are there trees whose levels are infinite that does not have a path?
Does anyone know an example of a tree that meets these characteristics?
Consider a rooted tree whose root has one child for each natural number. Child $n$ is the first node of a path of length $n$. That is, Child $0$ has no children, Child one has one child with no children, and so on.
The tree is infinite, and in fact has infinitely many levels with infinitely many nodes, but each path through the tree is finite.