So I've been currently studying this book on high dimensional probability by Roman Vershynin, which I find pretty awesome! However, I was wondering if in the first line of the proof of Theorem 3.1.1 (P. 43), where he proved a concentration inequality for high dimensional random vectors with independent components, it should be "it follows that" instead of "for simplicity, we can assume that"? This is because, since he assumed that $EX_i^2=1 \forall i,$:
$$2 \ge E[e^\frac{{X_i^2}}{||X_i||_{\psi_2}^2}] \ge E[1 + \frac{{X_i^2}}{||X_i||_{\psi_2}^2}] = 1 + \frac{{EX_i^2}}{||X_i||_{\psi_2}^2} = 1 + \frac{1}{||X_i||_{\psi_2}^2},$$ implying ${||X_i||_{\psi_2}^2} \ge 1 \forall i$.
Just checking to make sure, thanks!