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Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector.

Note that a random vector $x\in\mathbb{R}^n$ is sub-gaussian if for any $a\in\mathbb{R}^n$, $a\neq0$, we have $x^Ta$ is a sub-gaussian random variable. Additionally, a random vector $x\in\mathbb{R}^n$ is isotropic if $Ex=0$ and $\operatorname{Cov}(X)=I_n$. Equivalently, $x$ is isotropic if $$E[(x^Ta)]=\|\ a \|_2^2 \quad \quad \forall a\in\mathbb{R}^n.$$ Let $x$ be an isotropic sub-gaussian random vector and let $y$ be a sub-vector of $x$, so that if $x=(x_1,\dots,x_n)$ then $y=(y_{J_1},\dots,y_{J_k})$, where $J=\left\{J_1,\dots,J_k\right\}\subset\left\{1,\dots,n\right\}$.

I'm having trouble arriving to the desired result.

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