This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our introductory analysis course). It was proposed to him by another mathematician at Cambridge who taught him then, and is still there now, who had this problem on his list of "problems for enthusiasts". (http://en.wikipedia.org/wiki/Timothy_Gowers is the lecturer).
Let $z_i \in \mathbb{C}$ be a series of complex numbers with $\displaystyle \lim_{i \to \infty}{|z_i|}=0$. Show that we can choose signs $\alpha_i \in \left\{ {-1,1} \right\}$ such that $\displaystyle \sum_{i=1}^{\infty} \alpha_iz_i$ converges.
He deemed it fruitful to first try and prove the following weaker statement to get an idea of what's involved, yet I have not managed this either.
Let $z_i \in \mathbb{C}$ be a series of complex numbers with $|z_i|=1$. Show that we can choose signs $\alpha_i \in \left\{ {-1,1} \right\}$ such that $\displaystyle \sum_{i=1}^{\infty} \alpha_iz_i$ remains bounded. (In fact, it would seem that we can make $|\displaystyle \sum_{i=1}^{k} \alpha_iz_i| \leq \sqrt2 \ \ \forall k$).
Any ideas, thoughts, or solutions would be greatly appreciated.