Norms of Gaussian vectors: $\mathbb E\|h_N\|_2-\mathbb E\|g_n\|_2\ge\sqrt N-\sqrt n$ for $1\le n\le N$

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Norms of Gaussian vectors: $\mathbb E\|h_N\|_2-\mathbb E\|g_n\|_2\ge\sqrt N-\sqrt n$ for $1\le n\le N$ · Prove2Me