(When I write integrable I mean Riemann-integrable)
Let $A \subseteq \mathbb{R}^m$ be a block, $f_n:A\rightarrow R$ be integrable functions and $f_n \rightarrow_{un} f$. Proof that $f$ is integrable and $\int_A f_n \rightarrow \int_A f$
I tried to use here the Lebesgue Criterion. (http://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/Lebesgue%20Criterion%20for%20Riemann%20Integrability.pdf)
Since each $f_i$ is integrable, the set of discontinuities of each $f_i$ has measure $0$. Since the convergence is uniform, the set of discontinuities of $f$ will also have measure $0$, then $f$ is integrable. My "proof" is just a sketch, I believe it's needing formalization, but I'm not sure how to do it. I'm also not sure how to analyse the sequence of the integrals either.
Can someone help me? Thanks.
Remember that: $f$ is integrable on $A$ iff, for every $\varepsilon>0$, there exists a partition $P$ of $A$, such that $$ U(f,P)-L(f,P)<\varepsilon $$ where $L(f,P)$, $U(f,P)$ are the lower and upper sums of $f$ corresponding to $P$.
Let now $\varepsilon>0$. Since $f_n\to f$ uniformly on $A$, there exists an $N\in\mathbb N$, such that $n\ge N$, implies that $$ \sup_{x\in A}\lvert\, f_n(x)-f(x)\rvert<\frac{\varepsilon}{2(\mu(A)+1)}, $$ where $\mu(A)$ is the volume of $A$. Let $P$ a partition of $A$ for which $U(f_N,P)-U(f_N,P)<\frac{\varepsilon}{2}$. Note that, the partition $P$ divides $A$ in $K$ sub-blocks $\varDelta A_1,\ldots,\varDelta A_K$ of volumes $\mu(\varDelta A_1),\ldots,\mu(\varDelta A_K)$, respectively, and set $M_j(\,f_N),m_j(\,f_N)$ be the supremum and infimum of $f_N$ in $A_j$. Then $$ L(\,f_N,P)=\sum_{j=1}^K \mu(A_j)\,m_j(\,f_N), \quad U(\,f_N,P)=\sum_{j=1}^K \mu(A_j)\,M_j(\,f_N) $$ and $$ \lvert m_j(f_N)-m_j(f)\rvert < \frac{\varepsilon}{2(\mu(A)+1)},\,\,\, \lvert M_j(f_N)-M_j(f)\rvert < \frac{\varepsilon}{2(\mu(A)+1)} $$ and hence $$ U(f,P)-L(f,P)\le \cdots \le \frac{\varepsilon}{2}+\frac{\varepsilon \mu(A)}{2(\mu(A)+1)}<\varepsilon. $$ ETC