Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real-valued random variables. In a proof of a theorem, the author starts with
$$\lvert Cov(X_i,X_j)\rvert\leq \int_0^{\alpha_{\lvert i-j\rvert}}[Q_i(u)]^2+[Q_j(u)]^2du\quad (a)$$
where $(\alpha_{k})_{k\geq 0}$ is a nonnegative and nonincreasing sequence with $\alpha_0=1/2$ and $Q_k(u)=\inf\{x\in\mathbb{R}:P(\lvert X_k\rvert>x)\leq u\}$, the quantile function of $\lvert X_k\rvert$ (the quantile function of a random variable is nonincreasing). This inequality $(a)$ is true, and assume it holds. But from $(a)$, he inferred that
$$\sum_{i=1}^n\sum_{j=1}^n\lvert Cov(X_i,X_j)\rvert\leq 2\sum_{i=1}^n\int_0^1 \sum_{j=1}^n I({u<\alpha_{\lvert i-j\rvert}})[Q_i(u)]^2du \quad (b)$$
with $I(\cdot)$ denoting the indicator function.
I'm struggling to see $(a)\to (b)$. Can you see the reasoning of this?
My attempt
It's clear that $\int_0^{\alpha_{\lvert i-j\rvert}}[Q_i^2(u)+Q_j^2(u)]du=\int_0^1 I(u<{\alpha_{\lvert i-j\rvert}})[Q_i^2(u)+Q_j^2(u)]du$. Then $$\sum_{i=1}^n\sum_{j=1}^n\lvert Cov(X_i,X_j)\rvert\leq \sum_{i=1}^n\int_0^1 \sum_{j=1}^n I({u<\alpha_{\lvert i-j\rvert}})[Q_i^2(u)+Q_j^2(u)]du.$$
From this, I need to show that, in the summand, $Q_j^2(u)\leq Q_i^2(u)$ for each $1\leq i,j\leq n$, which is not clear to me. I see that the region of integration is maximized when $i=j$, which is equal to $[0,1/2]$. But I don't know the behavior of the height $Q_i^2(u)+Q_j^2(u)$ along $i,j$.
Reference: Rio's Book, pp. 9, Proof of Corollary 1.1
\begin{align} \sum_{i,j}|\operatorname{Cov}(X_i,X_j)| &\le \sum_{i,j}\int_0^1 1\{u< \alpha_{|i-j|}\}Q_i^2(u)\, du \\ &\quad+\sum_{i,j}\int_0^1 1\{u< \alpha_{|i-j|}\}Q_j^2(u)\, du \\ &=\sum_i\int_0^1 \sum_j 1\{u< \alpha_{|i-j|}\}Q_i^2(u)\, du \\ &\quad+\sum_j\int_0^1 \sum_i 1\{u< \alpha_{|i-j|}\}Q_j^2(u)\, du \\ &=2\sum_i\int_0^1 \sum_j 1\{u< \alpha_{|i-j|}\}Q_i^2(u)\, du. \end{align}