I have begun reading about the Haar function representation of Brownian motion and am trying to understand the proof of why the representation defines a Brownian motion. Specifically, I am looking at the proof of continuity. The proof I am reading over relies on two lemmas. The first states:
Lemma 1: There exist $M(\omega)<\infty$ so that $$P\bigg[\sup \frac{|Z_n(\omega)|}{\sqrt{\log n}} \leq M(\omega)\bigg] = 1$$ where $Z_n \sim N(0,1)$.
I understand this lemma and the proof of it. My confusion comes with the second lemma.
Lemma 2: $$\bigg| \sum_{k=0}^{2^j-1} Z_{2^j +k}(\omega) \int_0^t \psi_{2^j+k}(s) ds\bigg| \leq M_1(\omega) \sqrt{j} 2^{-j/2}$$ where $M_1(\omega) := 2M(\omega)\sqrt{2\log 2}$.
I have read two versions of this proof that relies on similar facts that I am confused with. The first proof stated that $$\sum_{0\leq k\leq 2^j-1} \int_0^t \psi_{jk}(s) ds \leq 2^{-j/2}.$$
The second essential stated the same thing by saying that, for each $t\geq 0$ and $j\in \mathbb{N}$, there exists only one $k$ such that $$\int_0^t \psi_{jk} (s) ds \neq 0.$$
My confusion comes with the fact that there exists only one such $k$ that leads to the integral being nonzero. Why is this true? Thank you for any help you can provide.