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I am looking for a sequence $\beta_n\in(0,1)$ such that

(i) $~ \prod_{n\in\mathbb N} \beta_n =0$,

(ii) $~\sum_{n\in\mathbb N} (1-\beta_n)< +\infty$.

Does such a sequence exist?

edit: i have changed to $\beta_n\in(0,1)$ instead of $\beta_n\in[0,1]$.

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    @pritam that is correct. As soon as a_n\le \theta<1 for all n>N_0 you get convergence to zero. For this reason this case is, by definition, excluded if one talks about convergent infinite products. This is why Cocopuffs wrote 'by convetion'.2012-06-30

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Assume without loss of generality when (ii) holds that $\beta_n\geqslant1-\frac12\log2$ for every $n\in\mathbb N$. Then, for every $n\in\mathbb N$, $\beta_n\geqslant\mathrm e^{-2(1-\beta_n)}$, hence $ \prod_{n\in\mathbb N}\beta_n\geqslant\exp\left(-2\sum_{n\in\mathbb N}(1-\beta_n)\right). $ Then (ii) implies that the RHS is positive hence the LHS is positive and (i) cannot hold.

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    thanks, I have also found this [link](http://math.stackexchange.com/questions/158089/infinite-products-reference-needed), which deals with my question and more.2012-07-01