A Diagonlization Argument on Trees
This argument was used in my most recent paper, and it arose by looking for a counterexample for a tree that does not witness a discrete $(0,1)$-generator. Taking a step back, I realized that it is simply a classical diagonilization argument. Instead of choosing numbers in the sequences of all ``listed’’ real numbers, we are choosing successors points at elements in the complete $\omega$-ary tree at every successor stage.
Background
The classical Cantor diagonlization argument proceeds roughly as follows. First, assume that the real numbers in the interval $(0,1)$ are countable, and simply list them out with their decimal expansion. Then, you construct a new real umber $x$ by selecting a different digit for the $n$th decimal place from the $n$th digit of the $n$th number. This new number $x$ will be an element of the interval $(0,1)$, and will be different from every number that is listed. This is an extremely low level overview of the argument as there is some care needed to ensure that we do not end up with an infini