Prove that $m$ is regular.

Question

For an interval $I=\{(x_1,x_2,…,x_p)|a_i≤x_i≤b_i (i=1,2,…,p)\}$, we define a set function $$m(I)=\prod_{i=1}^p(b_i-a_i)),$$ and $$m(I_1∪…∪I_n )=m(I_1 )+⋯+m(I_n)$$ whenever the intervals are pairwise disjoint. (Note: For p=1,2,3, then m is length, area, and volume respectively.)

Intervals are in the context of rudin Principles of Mathematical Analysis ch 11

Question Prove that $m$ is regular.

My work

Let $A$ be an interval.

Given $\epsilon >0$, choosing $$G=(a-\epsilon/2, b+\epsilon/2)=(a_1-\epsilon/2,b_1+\epsilon/2)\times (a_2-\epsilon/2,b_2+\epsilon/2)\times\ldots\times(a_p-\epsilon/2,b_p-\epsilon/2)$$ and $F=[a+\epsilon /2, b-\epsilon /2]$ Then, $$m(G)=\prod_{i=1}^pb_i-a_i-\epsilon$$ and $$m(F)=\prod_{i=1}^pb_i-a_i+\epsilon$$

After that,Am I getting, $m(G)-\epsilon\leq m(A)\leq m(F)+\epsilon$

Clear this doubt..