If $\prod _{n = 1 }^{\infty} a_n = 0$, then $lim_{n \rightarrow \infty} a_n = 0$ or $\exists n$ : $a_n =0$?

Question

I need to prove above(or contraposition) using $\epsilon - \delta$ definition. How can I do?

If $\prod _{n = 1 }^{\infty} a_n = 0$, then $lim_{n \rightarrow \infty} a_n = 0$ or $\exists n$ : $a_n =0$?

Answer 1

Another example is $$\prod_{n=1}^\infty(1-\frac1n)=\prod_{n=1}^\infty\frac{n-1}n=\frac12·\frac23·\frac34·\frac45···,$$ so that the factors converge to $1$.

There is a result that states that $$ \prod_{k=1}^\infty(1-b_k)= 0\text{ where }b_k\in(0,1) $$ if and only if $$ \sum_{k=1}^\infty b_k=\infty $$