Eq. (1): $\Pr\{\sum_j\eta_jp_j > \Omega\sqrt{\sum_jp_j^2}\}\le\exp\{-\Omega^2/2\}$ for independent symmetric $[-1,1]$ variables

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Eq. (1): $\Pr\{\sum_j\eta_jp_j > \Omega\sqrt{\sum_jp_j^2}\}\le\exp\{-\Omega^2/2\}$ for independent symmetric $[-1,1]$ variables · Prove2Me