Good evening to everybody. I am trying to understand a theorem which I found on John Conway's textbook ''Functions of one complex variable''. It is lemma $5.8$ on page 167, on the chapter of Weierstrass factorization theorem. The Lemma states the following :
'' Let $(X,d)$ a compact metric space and a sequence $\{g_n\}$ of continuous functions from $X$ to $\mathbb{C}$ such that the series $\sum g_n(x)$ converges absolutely and uniformly on $x\in X$. Then the product $f(x)=\prod_{n=1}^{\infty} (1+g_n(x))$ converges absolutely and uniformly on $X$. ''
For the definition of the absolute convergence of an infinite product $\prod_{n=1}^{\infty} z_n $, he has defined that the sum $\sum_n | \log(z_n) |$ has to converge. But I cannot see how he defines the uniform convergence of an arbitrary product of functions $\prod_{n=1}^{\infty} h_n(x)$ . So let us suppose (and I think in the proof he does that) that we must define it as the uniform convergence of the sum $\sum_n \log(h_n(x)) $. In the proof of the above Lemma, he says that since the series $\sum g_n(x)$ converges uniformly on $x\in X$ we can find an $n_0$ such that it holds $|g_n(x)| <1/2$ , for all $x\in X$ and for all $n\geq n_0$. This implies that $$ (*) \qquad \qquad \ \ \ \ \ \ \ \ \ | \log(1+g_n(x))| \leq \frac{3}{2} |g_n(x)|\qquad \qquad \qquad \qquad $$ for all $x$ and $n\geq n_0$ , since $\frac{1}{2} |z|\leq |\log(1+z)| \leq \frac{3}{2} |z|$for all $|z|<1/2$. Here comes now my problem. He says that we deduce that $$h(x)= \sum_{n=n_0}^{\infty} \log(1+g_n(x))$$ converges uniformly on $x\in X$. How is this true?? To prove the uniform convergence of this series, shouldn't we take the partial sums, and then try to bound them by something convergent and then use the Weierstrass M-test? I cannot see how the relation $(*)$ gives us what we need. If we try to take the partial sums under the $\| \|_{\infty}$ norm of the series $h$, then we get something like $$ | \log(1+g_m(x))+...+\log(1+g_n(x))|$$ which of course turns into the logarithm of a product, but how can we use now relation $(*)$ ? Any ideas ?