The proof of $\chi_A:I\to \mathbb R$ is integrable $\iff $ $\chi_A:J\to \mathbb R$ is integrable. I'm studying the integration on Jordan measurable sets.
Let $A\subset \mathbb R^n$ be bounded and $\chi_A$ be a characteristic function of $A$
Then, I want to prove that if $I\subset \mathbb{R^n}$ and $J\subset \mathbb{R^n}$ are intervals s.t. $A\subset I \subset J,$ then
\begin{align}
\chi_A: I\to \mathbb R \ \mathrm{is \ Riemann \ integrable \ on\ } I \iff  \chi_A : J\to \mathbb R \mathrm{\ is\ Riemann \ integrable\ on\ } J
\end{align}
For the proof of this, I think the theorem below is useful.
Theorem
Let $I\subset \mathbb{R^n}$ be a interval and $f:I\to \mathbb R$ be bounded.
$f$ is integrable on $I$ if and only if for
all $\epsilon>0$, there exists a partition of $I$ s.t. $S(f,\Delta)-s(f,\Delta)<\epsilon$ where $S$ is the upper sum and $s$ is the lower sum.

Suppose $\chi_A : I\to \mathbb R$ is integrable on I.
Let $\epsilon>0.$
Then, from the integrability of $\chi_A: I\to \mathbb R$, there exists the partition of $I$, $\Delta=\{I_k\}_{k=1}^M$ s.t. $S(\chi_A, \Delta)-s(\chi_A, \Delta)<\epsilon.$
Then, I have to find the partition of $J$ s.t. $S(\chi_A, \Delta')-s(\chi_A, \Delta')<\epsilon \  ($or $a \cdot \epsilon \ (a>0)).$
I'm having difficulty in finding such $\Delta'.$
I'd like you to gime me any help.
(This is the step of defining the integral on Jordan measurable sets.)

Just so you know, the definition of the integrability of functions on a interval is here.
Let $I\subset \mathbb R^n$ be an interval.
$f:I\to \mathbb R$ is integrable on $I$ $\underset{\mathrm{def}}\iff$ $\overline{\displaystyle\int_I} f(x) dx=\underline{\displaystyle\int_I}f(x) dx,$ where $\overline{\displaystyle\int_I}$ is upper integral and $\underline{\displaystyle\int_I}$ is lower integral.
 A: A partition $P$ of $J$ can be refined using the endpoints of $I$ to produce a partition $P'$. We then have $P' = P_1 \cup P_2$ where the interior of any subinterval in $P_1$ is disjoint from $I$ and the interior of any subinterval in $P_2$ is contained in $I$. Moreover $P_2$ is a partition of $I$.
We can decompose upper and lower Darboux sums as
$$S(\chi_A,P') =  S(\chi_A,P_1) +S(\chi_A,P_2), \quad  s(\chi_A,P') =  s(\chi_A,P_1) +s(\chi_A,P_2)$$
If $\chi_A$ is Riemann integrable on $J$, then given $\epsilon > 0$ there exists a partition $P$ such that $$S(\chi_A,P) - s(\chi_A,P) < \epsilon$$
Since $P'$ is a refinement of $P$, we have $s(\chi_A,P) \leqslant s(\chi_A,P')\leqslant S(\chi_A,P')\leqslant S(\chi_A,P)$
Note that a difference of upper and lower Darboux sums is always nonnegative, and, hence,
$$S(\chi_A,P_2) - s(\chi_A,P_2) \leqslant S(\chi_A,P_1) - s(\chi_A,P_1) + S(\chi_A,P_2) - s(\chi_A,P_2)\\ = S(\chi_A,P') - s(\chi_A,P') \leqslant S(\chi_A,P) - s(\chi_A,P) < \epsilon$$
Therefore, $\chi_A$ is Riemann integrable on $I$ (since $P_2$ is a partition of $I$ satisfying the Riemann criterion ).
Try for yourself to prove that integrability on $I$ implies integrability on $J$.
