clarification for 4.1.5 (dyadic property)

Stream: Carleson

[Edward van de Meent]

could someone take a look at the proof for 4.1.5? i'm struggeling to make sense of the steps after the proof of the base case where $l = k$. i've managed to eliminate the case where $l < k$ and $x \in X_k$, but i'm having trouble with the opposite case, and it is not at all as simple as expanding definitions, as the proof seems to suggest... another odd point of this proof is that it seems to want to do three cases, $l = k$, $l = k-1$, and $l < k-1$, but it doesn't mention the last assumption at all in the logic of that case...

[Edward van de Meent]

nevermind, i think i figured out how to prove the statement. However, if allowed, i'd like to adapt the proof in the blueprint to more clearly reflect the logic used.