let there be $\{z_1 ,..., z_n\}$ a group of complex numbers. Show that there's a subset $J \subset \{1,...n\}$ so that $$\lvert \sum_{k \in J}z_k \rvert \ge \frac{1}{4\sqrt2}\sum_{i=1}^n\lvert z_i \rvert$$
I've tried playing around with it, like squaring it and I got up to $$2^5 \sum_{k \in J}\lvert z_k \rvert ^2 -\sum_{i=1}^{n}\lvert z_i \rvert^2 \ge \sum_{i \neq k \in J} \lvert z_i \rvert \lvert z_k \rvert -2^6Re \sum_{i \neq k \in J} z_i \bar z_k$$
and here is where I got stuck
