It's Exercise 2.5.10 in High Dimensional Probability by Vershynin.
Exercise 2.5.10: (Maximum of sub-gaussians). ☕️☕️☕️ Let $X_1, X_2, \dotsc,$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that $$ \mathbb{E}\max_{i} \frac{|X_i|}{\sqrt{1+\log i}} \le CK,$$ where $K = \max_i \|X\|_{\psi_2}$. Deduce that for every $N\ge 2$ we have $$ \mathbb{E}\max_{i\le N} |X_i| \le CK\sqrt{\log N}.$$
I am able to show that $$ \mathbb{E} \max_i \frac{|X_i|}{\sqrt{1+\log i}} \leq CK$$
But when I try to deduce that
$$ \text{For every }N\geq 2, \mathbb{E}\max_{i\leq N} |X_i| \leq CK\sqrt{\log N}$$ I can only show the above inequality for $1\leq i \leq \left \lfloor \frac{N}{e} \right \rfloor $ since $ \sqrt{1+\log i}\leq \sqrt{\log N} \text{ in this range}.$ But for every $N\geq 2$, I cannot find a way to bound it.