Suppose we have two sequences $a_{i}$ and $b_{i}$ such that $a_{i}\in(0,\frac{1}{2})$ and $b_{i}\in (\frac{1}{2},1)$ and $\lim_{n\to \infty} a_{i}=\lim_{n\to \infty} b_{i}=\frac{1}{2}$. I am trying to find conditions under which $2^{2n}\prod_{i=1}^{n}a_{i}b_{i}=0$.
Any help is much appreciated.