Hoeffding to bound Orlicz norm

I have been reading from Weak Convergence and Empirical Processes, and came across the following: Let $ a_1,\ldots,a_n$ be constants and $ \epsilon_1,\ldots,\epsilon_n\sim$ Rademacher. Then

$ \mathbb{P}\left(\left|\sum_i\epsilon_i a_i\right|>x\right)\leq 2\exp\left(-\frac{x^2}{2||a||^2_2}\right)$

Consequently, $ ||\sum_i\epsilon_ia_i||_{\Psi_2}\leq\sqrt{6}||a||_2$ .

How does this follow (relation between Orlicz norm of Rademacher average and L2 norm of constants)? Thank you in advance for your time.