I was wondering if there is any proof that the limit of infinite product
$\lim_{n \to \infty} \prod_{i=1}^{n} x_i, \mathrm{where}$ $0 < x_i < 1$
is equal to 0 and that it does not converge to 1.
I have tried to prove that power series
$\sum_{n=1}^{\infty}\mathrm{log}(x_n)$
does converge, hence ensuring that the product does converge to anything other than 0, but I seem to have reached a dead spot.
Thank You for any tips/ideas/solutions.