This is my first question here and I have thought about it for a long time. I found following Lemma in a paper:
Consider the independent random variables $X_1,\dots,X_M \sim \mathcal{N}(0,1)$. Then it holds \begin{align*} \mathbb{E}\left[\max_{j=1,\dots,M}X_j^2 \right]\leq 3\log M +1. \end{align*}
The proof uses \begin{align*} \max_{j=1,\dots,M}X_j^2\leq \sum_{j=1}^MX_j^2\cdot\chi\left(\lvert X_j\rvert>T\right)+T^2 \end{align*} for any $T>0$. At the beginning one shows $\mathbb{E}\left[X_j^2\cdot\chi\left(\lvert X_j\rvert>T\right)\right]\leq \left(1+(2\pi)^{-1/2}T\right)e^{-T^2/2}$. The author uses \begin{align*} \mathbb{E}\left[X_j^2\cdot\chi\left(\lvert X_j\rvert>T\right)\right]=(2\pi)^{-1/2}\int_T^{\infty}x^2e^{-x^2/2}\mathrm{d}x. \end{align*} I think this is not correct, since the right hand side should be doubled. Although, I tried to follow the proof further but I am not able to fix this mistake.
The question is: Is this a mistake in the proof or am I overseeing something very trivial? And if the proof is wrong, does anyone know a proof or how to fix it?