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 $\beta_n\in[0,1]$.
$\endgroup$ 81 Answer
$\begingroup$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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