I'm studying infinite products appearing in complex analysis these days. There are lots of theorems regarding it, for example, for $0<u_n <1, \prod (1-u_n) > 0$, iff $\Sigma u_n < \infty $.
I come up with a following question which No elementary book say about it..
within above condition, $\prod u_n =1 $ is impossible, but i thought intuitively that by approaching$ u_n$ rapidly to 1, it is possible to get a value of infinite product is larger than 1-$\epsilon$ for any $\epsilon >0$ (of course our$ u_n $can depend on $\epsilon$) if anyone know about theorem regrading it or can prove about this? thanks.