Is it possible to define the class of functions $\phi: [1,\infty)\rightarrow[0,0.6)$ with infinite product which converges to $0$?

Question

Consider $\phi: [1,\infty)\rightarrow[0,c)$ for some constant $c=0.6$. I am wondering if there are any required properties of $\phi$ such that $$\prod_{i=0}^{\infty} (1-\phi(i))=0.$$

I assume defining asymptotics, such as Big O notation would be desirable. Any help would be much appreciated.

Answer 1

Let $f(x)=\ln(1-x)/\ln(0.4)$, which is a bijection $f\colon[0,0.6)\to[0,1)$. Now, letting $\psi(i)=f(\phi(i))=\ln(1-\phi(i))/\ln(0.4)$, you see that $\prod_{i=0}^\infty(1-\phi(i))$ converging to $0$ is equivalent to $\sum_{i=0}^\infty\psi(i)$ diverging to $\infty$, with $\psi(i)\in[0,1)$.