If $\boldsymbol{x}\sim\mathcal{N}(0,\boldsymbol{I}_n)$, then the expected value of the Euclidean norm has the following lower bound: $$ \mathbb{E}[\|\boldsymbol{x}\|_2]=\mathbb{E}\left[\sqrt{\boldsymbol{x}^T\boldsymbol{x}}\right]\geq\sqrt{n}(1-o(1)). $$ I was wondering how to get this lower bound. In the book I have read, the author mentioned that the lower bound can be obtained via the concentration inequality of chi-square random variable.

Thank you in advance.


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