I came across the following inequality while reading Patterson's book, "An Introduction to the theory of Riemann zeta function" on Page 107:
For $ |t^{2/3}|<X\leq|t|$ then
$$\left|\sum_{n\leq X}n^{-1/2-it}\right|\leq \sum_{j< \frac{\log X}{\log 2}}\left|\sum_{2^{-j-1}X<n<2^{-j}X}n^{-1/2-it}\right|$$
Why is the above inequality (especially the indices on right summation) true? So far I have come across the term "dyadic summation" but haven't been able to find a resource that outlines this technique. I appreciate any resources that prove the above inequality and also a reference for the dyadic summation technique