Products of sub-Gaussians are sub-exponential

ProofAdvanced⏱ 30 minHigh-Dim Probabilityproductsub-exponentialyoung inequality

Problem

Let $X,Y$ be sub-Gaussian (not necessarily independent). Prove that $XY$ is sub-exponential with

\|XY\|_{\psi_1} \le \|X\|_{\psi_2}\|Y\|_{\psi_2}.

Write a complete argument. Every inequality you use should be one you could state precisely.

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Problem	Field	Type	Difficulty	Time
The psi-2 norm of a Rademacher sign	High-Dim Probability	Numerical	Beginner	15 min
A variable with all moments finite that is not sub-Gaussian	High-Dim Probability	Proof	Intermediate	30 min
The psi-2 norm of a Gaussian	High-Dim Probability	Numerical	Intermediate	15 min
From moment growth to a tail bound	High-Dim Probability	Proof	Advanced	1 hour