# Tail bounds for maximum of sub-Gaussian random variables

I have a question similar to this one, but am considering sub-Guassian random variables instead of Gaussian. Let $X_1,\ldots,X_n$ be centered $1$-sub-Gaussian random variables (i.e. $\mathbb{E} e^{\lambda X_i} \le e^{\lambda^2 /2}$), not necessarily independent. I am familiar with the bound $\mathbb{E} \max_i |X_i| \le \sqrt{2 \log (2n)}$, but am looking for an outline of a tail bound for the maximum.

A union bound would give $$\mathbb{P}(\max_i |X_i| > t) \le \sum_i \mathbb{P}(|X_i| > t) \le 2n e^{-t^2/2},$$ but I am looking for a proof of something of the form $$\mathbb{P}(\max_i |X_i| > \sqrt{2 \log (2n)} + t) \le \mathbb{P}(\max_i |X_i| > \mathbb{E} \max_i |X_i| + t) \le 2e^{-t^2/2}.$$ Does anyone have any hints?