This is from Palka's An Introduction to Complex Function Theory, Chapter 5, lemma 1.1:
If a function $f$ is analytic in an open set $U$, then $\int\limits_{\partial R}f(z)dz = 0$ for every closed rectangle $R$ in $U$.
My specific question is how do we know what Palka claims about the diagonal and perimeter of the divided subrectangles when the discussion has been around the integral of an analytic function $f$ over the boundaries of rectangles.
Palka begins his proof by first arguing why given a closed rectangle $R$ in $U$, one can subdivide it into four congruent rectangles $R_1^{i}, i = 1,2,3,4$ such that if $I = \int\limits_{\partial R}f(z)dz$ then for some $i = 1,2,3,4$ we have that $|I^i_1|\geq 4^{-1}|I|$, where $I^i_1 = \int_{\partial R_1^i}f(z)dz$. Then by induction Palka argues that we can divide the original rectangle $R$ even further so that at the $n$th step we have $|I_n| \geq 4^{-n}|I|$, where the subscript has been removed (as we've already argued about the existence in terms of the four starting subrectangles).
Then, Palka states that
Let $d_n$ designate the length of the diagonal of the rectangle $R_n$, and let $L_n$ be its perimeter. If $d$ and $L$ represent the corresponding quantities for the original rectangle $R$, then by construction $d_n = 2^{-n}d, L_n = 2^{-n}L$.
What I don't understand is that what is the construction he is talking about, in the sense that to this point the chain of subrectangles has been formed with a condition on the integral of $f$ over the boundaries of the original and subrectangles. Therefore it is unclear to me how we know that at each step of the division procedure, the diagonal and perimeter of the original rectangle are halved.