Positivity of the Gaussian absolute first moment

The Gaussian absolute first moment $C = \int_{\mathbb{R}} |t|\,dN(0,1)(t)$ is strictly positive. Proof by contradiction: if $C = 0$ then $N(0,1)$ is supported at $\{0\}$, forcing its variance to $0$, contradicting $\mathrm{Var}(N(0,1)) = 1$.