Let $\phi:[1,\infty)\rightarrow (0,1)$. Is $\lim_{n\rightarrow\infty}\prod^n_{i=1}\phi(i)=0$? If so, could this be proven? I specifically have the function \begin{equation} \phi(m)=1-\frac{2\lfloor\log_2(f(m))\rfloor}{f(m)}\cdot \frac{F_{2m+1}}{F_{2m+2}} \end{equation}
where $f(m)=\frac{3^{m}\left(n+1\right)}{2^{m}}-1$ such that $n\in\{1,2,3,4,...\}$ and $F_{\zeta}$ is the Fibonacci number at $\zeta$. Instead of dealing which a complicated infinite product, it would be desirable to answer the first queation (i.e. with a general function). I have not specified whether $\phi$ is an injection or surjection (but if it makes the problem easier, I would lean towards an injection). Any help is much appreciated.