SCHWEITZER 2008 Problem 11

Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$ . Prove that $$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$ (translated by Miklós Maróti)