I need help in trying to understand this proof (at the beginning of page 23, page 16 of the pdf) : https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch2.pdf#page26
I dont understand this part : Let $S_i$ be an rectangle whose interior ${S_i}^◦$ contains $R_i$ such that $μ(S_i) ≤ μ(R_i) + \frac{\epsilon}{2^{i+1}}$
If the interior of the rectangle S contains the rectangle R, how its possible that the measure of R is bigger or equal than the measure of S? I am misunderstanding something here?
Thanks.
Edit : Now I realize, the union of $S_i$ is bigger than the union of $R_i$, but each $S_i$ is less or equal than each $R_i$, is that correct?