I'm trying to show that if $S = \{v_1, \dots, v_q\}$ is orthonormal, then for every $v$ in a vector space $V$ we have
$$ \sum_{k=1}^q|\alpha_k|^2 \leq \Vert v \Vert_V^2 $$
where $\alpha_k = \langle v_k, v\rangle$.
Using Cauchy-Schwartz, I was able to show $\sum_{k=1}^q|\alpha_k|^2 \leq q\Vert v \Vert^2$, but that doesn't seem helpful. I have a hint to consider $w = \sum_{k=1}^q\alpha_kv_k$ and $\langle v, w \rangle$, but it's not clear to me how to use those.