$\int_{K_n}e^{-x^2-y^2}=\int_{-n}^n e^{-x^2}dx\int_{-n}^n e^{-y^2}dy$ I stuck in understand a formula. Let a sequence $K$ in $\Bbb R^2$ be defined as $K_n:=[-n,n]\times [-n,n]$. Then $\int_{K_n}e^{-x^2-y^2}=\int_{-n}^n e^{-x^2}dx\int_{-n}^n e^{-y^2}dy$. Why does this hold?
 A: Let $\mathscr{P}^k$ and $\mathscr{Q}^k$ sequeces of Riemman's partitions of $[-n,+n]$ given by
$$
\mathscr{P}^k:=\left\{-n=x^k_0<x^k_{1}<x^k_2<\ldots<x^k_i<\ldots<x^k_k=+n
\right\}
\\
\mathscr{Q}^k:=\left\{-n=y^k_0<y^k_{1}<y^k_2<\ldots<y^k_j<\ldots<y^k_k=+n
\right\}
$$
whit
$ x^k_i=-n+\frac{i}{k}(2n)$ and $y^k_i=-n+\frac{j}{k}(2n)$. For a fixed $k$ and fixed $i$ set a $\tilde{x}^k_i\in I^k_i:=[x^{k}_{i-1},x^{k}_{i}]$. For a fixed $k$ and fixed $j$ set a $\tilde{y}^k_j\in J^k_j:= [y^{k}_{j-1},y^{k}_{j}]$. Then
$$
\int_{-n}^{+n} e^{-x^2}\mathrm{d} x= \lim_{\|\mathscr{P}^k\|\to 0}\sum_{I^k_i}e^{-(\tilde{x}^k_i)^2}\mathrm{lang}(I^k_i)
\quad \mathrm{ and } \quad 
\int_{-n}^{+n} e^{-y^2}\mathrm{d} y= \lim_{\|\mathscr{Q}^k\|\to 0}\sum_{J^k_j}e^{-(\tilde{y}^k_i)^2}\mathrm{lang}(J^k_j)
$$
Besides that
$$
\mathscr{R}^{k}=\left\{B_{ij}^{k} \;\left|\;
B_{ij}^{k}:= [x_{i-1}^{k},x_{i}^{k}]\times [y_{j-1}^{k},y_{j}^{k}]
,\quad 1\leq i,j\leq k
\right.\right\} 
$$
is a sequence of Riemman's partitions of $K_n$ such that $\|\mathscr{R}_k\|=\max\mathrm{diam}(B^{k}_{ij})\to 0$ if $k\to \infty$. 
Then the double integral 
$\int_{K_n}e^{-x^2-y^2}\mathrm{d}\,A $ is equal to limit of riemman sum 
\begin{align}
\int_{K_n}e^{-x^2-y^2}\mathrm{d}\,A 
=&
\lim_{k\to\infty}
\sum_{B^k_{ij}\in \mathscr{R}^k}
e^{-(\tilde{x}_i^k)^2-(\tilde{y}_j^k)^2}
\cdot 
\mathrm{area}(B^k_{ij})
\\
=&
\left(
\lim_{k\to\infty}
\sum_{I^k_{i}\in \mathscr{P}^k}
e^{-(\tilde{x}_i^k)^2}
\cdot 
\mathrm{lang}(I^k_{i})
\right)
\cdot
\left(
\lim_{k\to\infty}
\sum_{J^k_{j}\in \mathscr{Q}^k}
e^{-(\tilde{y}_j^k)^2}
\cdot 
\mathrm{lang}(J^k_{j})
\right)
\\
=&
\left(
\int_{-n}^{+n} e^{-x^2}\mathrm{d} x
\right)
\left(
\int_{-n}^{+n} e^{-y^2}\mathrm{d} y
\right)
\end{align}
A: This holds by applying Fubini's theorem or Tonelli's theorem for non-negative functions.
