Does the infinite product of the reciprocals of a decreasing (or increasing) function equal zero?

Question

I've made a short document explaining what I've just claimed. I'd like to know if the criteria for the infinite product to be zero is enough to hold for all decreasing and increasing functions. Theorem.

Answer 1

Infinite products are fun. Check out Rudin, Real and Complex Analysis (Theorem 15.5, page 322). There Rudin gives a clear (and simple) proof of the fact that an infinite product of "fractions" $0 < 1 - u_n \leq 1$ converges to a nonzero (positive) value exactly when the complementary sequence of fractions $0 ≤ u_n < 1$ is summable. In symbols, $$ \sum_{n=1}^{\infty} u_n < \infty \iff \prod_{n=1}^\infty (1-u_n) >0 .$$