I am reading this article about the Riemann Hypothesis and it states:
Lemma 1.Suppose ${ (a_n) }$ is a series. If $\sum_{n=1}^\infty a_n < \infty$ , then the product $\prod_{n=1}^\infty (1+ a_n)$ converges. Further, the product converges to 0 if and only if one if its factors is $0$.
I wonder how can one proof it? Also I am surprised about the fact that when one $a_n$ is $0$ that then the product converges to $0$. If I imagine that every $a_n$ is $0$, then I would assume that the product converges to $1$ ?
Take the logarithm of the product:
$$\log\left(\prod_{n=1}^\infty(1+a_n)\right)=\sum_{n=1}^\infty\log(1+a_n)$$
for all $\;n\;$ big enough, $\;|a_n|<\frac14\;$ , say, so the sereis above is a positive one and
$$\frac{\log(1+a_n)}{a_n}\xrightarrow[n\to\infty]{} 1$$
and the limit comparison test gives you the convergence
Can you finish now the argument?