I would like to show following statement:

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> For $M\geq 2,\ X\_1,\dots,X\_M\sim^{iid}\mathcal{N}(0,1)$ independent, it holds $P(\max\_{i=1,\dots,M}\lvert X\_i\rvert\geq y)\leq Me^{-y^2/2}$. 
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I think it is possible to show it via induction, but I have got problems at the start:

So, I want to show $P\left(\max(\lvert X\_1\rvert,\lvert X\_2\rvert)\geq y \right)\leq 2 e^{-y^2/2}$. One can show $P\left(\max(\lvert X\_1\rvert,\lvert X\_2\rvert )\geq y\right)=4\Phi(y)(1-\Phi(y))$ where $\Phi$ denotes the distribution function. I know that $1-\Phi(y)\leq e^{-y^2/2}$, so this inequality seems to be a little sharper to me. Maybe someone knows how to prove this or an alternative prooving strategy.