Pi Product Notation from some number up to $\infty$

Question

Does it mean when

$\displaystyle\prod_{x=b}^{\infty}f(x) = 0$,

there must be at least 1 number ( $a$ ) between $b$ and $\infty$ inclusive such that $f(a) = 0$?

Answer 1

No, it just means that

$$\lim_{B\to\infty} \prod_{x=b}^B f(x) = 0$$

which is true, for example, for the function $f$ which is constantly equal to $\frac{1}{2}$ for all $x$.

Answer 2

If a function $|f(x)|$ is less than $1$ when $x\in [a,\infty)$ then $$\prod_{x=b}^\infty f(x)=\begin{cases}0 & a\le b\\\prod_{x=b}^{a-1} f(x)\times\prod_{x=a}^\infty f(x)=\prod_{x=b}^{a-1} f(x)\times0=0 & a>b\end{cases}$$ so even if no $f(x)$ is $0$ the product can be equal $0$