# Bound for the maximum element of a normal random vector

I'm trying to understand the following bound (Meinshausen 2010):

Let $X$ be an $n\times p$ matrix with normalised columns; $||X_j||_2 = 1$ for $j \in \left\{ 1 \ldots p \right\}$ (we also assume that $p > 10$). Let $\varepsilon$ be a $p$-dimensional multivariate-normal random variable with distribution $\mathcal N(0,\sigma^2 I_p)$.

The author claims (without any further explanation) that:

$$\mathbb P\left[ \max_{1 \leq j \leq p} |X_j^T \varepsilon| \leq 2\sigma \sqrt{ \log(p) / n}\right] \geq 1 - 1/p$$

Is there a simple way to see this? What intermediate results are being used here?

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