Let $f_n$ uniformly convergent to$ f$. Proof that $f$ is integrable and $\int_A f_n \rightarrow \int_A f$ (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. 
 A: 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
