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This question is from Folland 1.32:

Suppose $\{a_j\}\subset(0,1)$.

a. $\prod(1-a_j)>0$ iff $\sum a_j<\infty$. (Compare $\sum\log(1-a_j)$ to $\sum a_j$).

b. Given $B\in(0,1)$, exhibit a sequence $a_j$ such that $\prod(1-a_j)=B$.

In particular, I would like to ask about item a. I have shown that if $\prod (1 - a_j) > 0$, then$\sum a_j < \infty$. However, I am having trouble to show if $\sum a_j < \infty$, then $\prod 1 - a_j$.

There is a particular identity that I have been trying to use but to no vail:

If $x \in (0, 1)$, then $x \leq \int_{1 - x} ^1 \frac{1}{t} \,dt = \ln\left(\frac{1}{1 - x}\right)\leq \frac{x}{1 - x}$.

I know similar questions have been asked, but I couldn't follow the implications written for this direction of the proof: Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$.

Mathematics_Beginner
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    Check this: https://math.stackexchange.com/q/2476326/42969, or this: https://math.stackexchange.com/q/950653/42969, or this: https://math.stackexchange.com/q/944678/42969, or this: https://math.stackexchange.com/q/695881/42969. – Martin R Feb 28 '22 at 08:01
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    As you can see, this has been asked and answered many times (found with Approach0. If none of those Q&As solves your problem then I would suggest that update your question and describe exactly what you do not understand in the given answers. – Martin R Feb 28 '22 at 08:05

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