If $a_n \in (0,1)$, then $x_n = (1- a_1) (1- a_2)... (1- a_n)$ is convergent only when $S = \displaystyle \sum^{\infty}_{n=1} a_n < \infty$

Question

Proof that if $a_n \in (0,1)$ then a sequence of products given by $$x_n = (1- a_1) (1- a_2)... (1- a_n)$$ is convergent to finite positive limit only when $S = \displaystyle \sum^{\infty}_{n=1} a_n < \infty$

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I tried to apply someting similar to:

If $a_1,\ldots ,a_n >0$ and $S=a_1+\ldots + a_n <1$, then $(1+a_1)(1+a_2)\ldots (1+a_n)(1-S) < 1$

$$(1-a_1)(1-a_2)\ldots(1-a_n)=1-(a_1+a_2+\ldots+a_n)+(a_1a_2+a_1a_3+\ldots+a_{n-1}a_n) - (a_1a_2a_3+a_1a_2a_4+\ldots)+ \ldots$$

So: \begin{align*} 1&=1\\ a_1+a_2+\ldots+x_n &= S \\ a_1a_2+a_1a_3+\ldots+a_{n-1}a_n &< S^2 \\ a_1a_2a_3+a_1a_2a_4+\ldots &< S^3 \\ &\vdots \\ a_1a_2a_3\ldots a_{n-1}+\ldots &< S^{n-1}\\ a_1a_2\ldots a_n &< S^n. \end{align*}

It follows that $$(1+a_1)(1+a_2)\ldots(1+a_n) < 1-S+S^2- \ldots $$

And I don't know what to do next but I have a feeling that this idea can be implied that case.