I am trying to understand the soundness bound reached in Theorem 4 of this paper, which deals with separating $QMA$ and $QCMA$ with respect to an in-place oracle. To state just the part I am confused with:
For some fixed integer $n$, let $N = 2^{{n}}$. Let $S \subseteq [N^{2}]$ be a set such that $|S| = N$. Let
$$A = \frac{1}{2^{n}}~~ \Big| \sum_{i \in S} \beta_{i} \Big|^{2}$$
and
$$B = \sum_{i \in Z_{even} \cap S} | \beta_{i} |^{2}.$$
where each $\beta_{i}$ is a complex number, $[n] = \{1, 2, \ldots, n \}$ and
$$\sum_{i \in [N^{2}]} | \beta_{i} |^{2} = 1.$$
$Z_{even}$ is the set of all even integers, where we know that $|Z_{even} \cap S| = \frac{1}{3} |S|$. I am trying to prove that
$$1 \geq \frac{\sqrt{3A} +(\sqrt{2} - 1)B}{\sqrt{2}}.$$
As per the paper, I am using the triangle and Cauchy Schwarz inequalities but I get stuck after a point. Here is my incomplete attempt at a solution:
$$A \leq \frac{1}{2^{n}}~ \Bigg(\sum_{i \in S} |\beta_{i}|\Bigg)^{2} \leq ~ \sum_{i \in S} |\beta_{i}|^{2} = B + \sum_{i \in Z_{odd} \cap S} | \beta_{i} |^{2}.$$
I am stuck after this.