Proof of The Standard Normal Densities in Closed Form

Routine facts about real numbers are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity).

Both formulas follow by substituting the value of the Gaussian constant, which is $c_{\mathrm{G}}=\sqrt{2\pi}$ by The Value of the Gaussian Integral (i).

On the real line. By the definition of the standard normal density (Standard Normal Distribution (i)),

$$\varphi(x)=c_{\mathrm{G}}^{-1}\exp(-x^{2}/2)\qquad(x\in\mathbb{R}),$$

and substituting $c_{\mathrm{G}}=\sqrt{2\pi}$ gives (i).

On $\mathbb{R}^{k}$. By the definition of the standard normal density on $\mathbb{R}^{k}$ (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (i)),

$$\varphi_{k}(x)=c_{\mathrm{G}}^{-k}\exp\bigl(-\lVert x\rVert^{2}/2\bigr)\qquad(x\in\mathbb{R}^{k}),$$

and $c_{\mathrm{G}}^{-k}=(\sqrt{2\pi})^{-k}=1/(\sqrt{2\pi})^{k}$, which gives (ii).