This is taken from Rudin's Real and Complex Analysis pg.314 I'm trying to understand the proof of Muntz Theorem in the text.
Let $\lambda_1, \lambda_2,..$ be positive real numbers such that $\sum\frac{1}{\lambda_k}<\infty$. Define the function $$f(z):=\frac{z}{(2+z)^3}\Pi_{k=1}^{\infty}\frac{\lambda_k-z}{2+\lambda_k+z}$$
Note that since
$$1-\frac{\lambda_k-z}{2+\lambda_k+z}=\frac{2+2z}{2+\lambda_k+z} (*)$$
The author claims that the infinite product that appears in the definition of f converges uniformly on compact subsets of $\mathbb{C}\backslash(\{-2\}\cup\{-2-\lambda_k\}_{k=0}^{\infty}$ and that each factor in the infinite product of f has absolute value less than 1 for $Re(z)>-1$
My question
1) What is the justification for the uniform convergence of the product
2) How can we prove that each factor has absolute value less than -1
For (1) there is a theorem on pg.300 that says
Suppose $f_n \in H(\Omega)$ for $n=1,2,3,...$, no $f_n$ is identically zero in any component of $\Omega$ and $$\sum|1-f_n(z)|$$ converges uniformly on compact subsets of $\Omega$. Then the product $$f(z)=\Pi_{n=1}^{\infty}f_n(z)$$ converges uniformly on compact subsets of $\Omega$.
So for (1) my real question is how (*) implies $\sum|1-f_n(z)|$ converges uniformly. This might have to do with the Weierstrass M-Test but I don't see how.
Any help would be appreciated