The following function is given: $ I_n=\int \limits_{R^{n}}^{}e^{-|x|^{2}}d\lambda^{n}(x) $
Prove that $$ I_n = (I_1)^{n} $$
To me this seems to be a rather trivial statement, because I think you can for example write the integral like this for n=2:
\begin{align} I_2=\int\limits_{\mathbb{R}^n}\exp(-\vec{x}\dot{}\vec{x})\mathrm{d}x=\prod_{i=1}^n \int\limits_{\mathbb{R}}\exp(-x_i^2)\mathrm{d}x_i=\prod_{i=1}^n\sqrt{\pi}=\pi^{\frac{n}{2}} \end{align}
Is there a formal proof of this statement?