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Proof of expectation of mean of normal distribution is 0

In one (pdf) copy of this passage the assertion instead is There exists $c_1\gt 0$ such that $f(x) = \max\{|x|,x^2\}e^{-x^2/2}\le c_1 e^{-c_1 |x|}$ for all $x\in \mathbb R.$ Here is a screen shot: ...
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