Let $f(z)$ be any continuous function defined in the complex plane with the property that $$\bigg|\int_{R_n}f(w)dw\bigg|\leq n^2\log(n),$$ for any $n>1$ and any square $R_n$ with side length $n$. Show that for any $n>5$, $$\bigg|\int_{R_n}f(w)dw\bigg|\leq n^2\log(n/2).$$
How do I show this? I know the proof is related to a proof of Cauchy's integral theorem for rectangles, in which the rectangle is divided up into smaller and smaller pieces, but this square gets larger as $n\to\infty$.