$\int_{a}^{+\infty}dx\int_{c}^{+\infty}f(x,y)dy=\iint_{D}f(x,y)dxdy$? Let $D:=[a,+\infty)\times[c,+\infty) $,where $a,c \in \mathbb{R};$  and let $f:D \mapsto \mathbb{R} $ be a nonnegative continuous function.
Prove that if the existence of either of the iterated integrals $\int_{a}^{+\infty}dx\int_{c}^{+\infty}f(x,y)dy$ and  $\int_{c}^{+\infty}dy\int_{a}^{+\infty}f(x,y)dx$ ,then the improper double integral $\iint_{D}f(x,y)dxdy$ converges to the value of the iterated integral in question.

WLOG,suppose $\int_{a}^{+\infty}dx\int_{c}^{+\infty}f(x,y)dy=I(\in\mathbb{R}).$
Given any $b,d\in\mathbb{R}$  with $b>a,d>c$,by Fubini's theorem,$$\iint_{R}f(x,y)dxdy=\int_{a}^{b}dx\int_{c}^{d}f(x,y)dy,R=[a,b]\times[c,d].$$
Let $\varphi(x,\beta ):=\int_{c}^{\beta}f(x,y)dy,$ then for each fixed $\beta_{0}(>c)$ and $\alpha^{"}>\alpha^{'}>a,$ we obtian $$\int_{a}^{\alpha^{"}}\varphi(x,\beta_{0} )dx\geq \int_{a}^{\alpha^{'}}\varphi(x,\beta_{0} )dx.$$
Since $\int_{a}^{+\infty}dx\int_{c}^{+\infty}f(x,y)dy=I,$
$\int_{a}^{\alpha}\varphi(x,\beta_{0} )dx$ with respect to $\alpha$ has upper unbound on $[a,+\infty).$
So $\lim_{\alpha \rightarrow +\infty}\int_{a}^{\alpha}\varphi(x,\beta_{0} )dx$ exists and is a real number . 
Next we have $$\iint_{[a,+\infty)\times[c,\beta_{0} ]}f(x,y)dxdy=\int_{a}^{+\infty}\varphi(x,\beta_{0} )dx.$$
At this time, If we  can prove  $$\lim_{\beta \rightarrow +\infty}\int_{a}^{+\infty}\varphi(x,\beta_{} )dx=\int_{a}^{+\infty}\left (  \lim_{\beta \rightarrow +\infty}\varphi(x,\beta_{} )\right )dx,\quad (*)$$then $$\int_{a}^{+\infty}dx\int_{c}^{+\infty}f(x,y)dy=\iint_{D}f(x,y)dxdy=I$$

But until now,I have no idea to prove $(*)$ is true.I need some help to deal 
with it ,or better solutions in other ways .
 A: $\forall \epsilon>0,\exists b>a$, s.t.$$\Biggl|\int_{a}^{b}dx\int_{c}^{+\infty}f(x,y)dy-I\Biggr|<\varepsilon$$, let $\{d_n\}$ be an ascending sequence satisfying $\lim_{n\to+\infty}d_n=+\infty$ and $\varphi_n(x)=\int_c^{d_n}f(x,y)dy$, $D_n=[a,b)\times[c,d_n)$, then
$$\int_{a}^{b}dx\int_{c}^{d_n}f(x,y)dy=\iint_{D_n}f(x,y)dxdy,$$
by Arzelà's dominated convergence theorem for the Riemann integral, 
$$\lim_{n\to\infty}\int_a^b \varphi_n(x)dx=\int_a^b \lim_{n\to\infty}\varphi_n(x)dx=\int_{a}^{b}dx\int_{c}^{+\infty}f(x,y)dy$$
Then therer exists $N>0$, s.t. $\forall n>N$,
$$\Biggl|\int_{a}^{b}dx\int_{c}^{d_n}f(x,y)dy-\int_{a}^{b}dx\int_{c}^{+\infty}f(x,y)dy\Biggr|<\epsilon,$$
the rest of the proof is obvious.
The idea to prove Arzelà's theorem is to find a increasing sequence of continuous functions $\{g_n\}$ satisfying $\lim_{n\to\infty}\varphi_n>g_n(x)>\varphi_n(x)$ such that $\int_a^b g_n(x)-\varphi_n(x)dx$ is small enough, then by Dini's theorem $\{g_n\}$ converges to $\lim_{n\to\infty}\varphi_n$ uniformly, thus $\lim_n\int \varphi_n=\int \lim_n \varphi_n$
