I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:

We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such that

(i) $|\Lambda_n|\leq6\cdot 2^n$, $n=1,2,\ldots$,

(ii) $|c_I|\leq C_1'2^{-n}$, $I\not\in\Lambda_n$,

where in (ii), $c_I$ is any of the three Haar coefficients associated to $I$. **It is easy to see that this implies (8.8).**

Here, (8.8) bounds the size of the $n$th largest Haar coefficient in terms of total variation: $$ \gamma_n(f)\leq 36C_1'\frac{V_Q(f)}{n}. $$ When they say (i) and (ii) implies (8.8), this is operating under the assumption that $V_Q(f)=1$. I assume the logic here is based on some well-known wavelet trick (like the Calderón–Zygmund decomposition), but I am not very familiar with this literature.

Why do (i) and (ii) together imply (8.8) when $V_Q(f)=1$?