I was reading a paper which seemed to use (without proof) several instances of the following relation: $$ \prod_{n = 0}^\infty \left(1 - \frac{1}{f(n) \cdot g(n)}\right)^{f(n)} > 1 - \sum_{n = 0}^\infty \frac{1}{g(n)}. $$ where $f, g\colon\mathbb{N} \to (1, \infty)$ and the right side is greater than $0$.
Is this in fact always true? If so is there a straightforward proof of it? If not are there straightforward conditions which guarantee it holds?
$$\prod_{n=0}^\infty\left(1-\frac1{f(n)g(n)}\right)^{f(n)}>\prod_{n=0}^\infty\left(1-\frac1{g(n)}\right)>1-\sum_{n=0}^\infty\frac1{g(n)}$$ where the first inequality is basically Bernoulli's, and the second one follows from $$\prod_{n=0}^N(1-a_n)>1-\sum_{n=0}^N a_n\qquad(N>0, 0<a_n<1)$$ which is easy to show using induction (on $N$).