Let $a_n = 1-\frac{1}{n^2}, g_n (z)= \frac{a_n -z}{1-a_n z}= 1-\frac{1}{n^2}\frac{1+z}{1-a_n z} $
$$ f(z)= \Pi_{n=1}^{\infty} g_n(z) $$
Show that $f(z) $ is holomorphic and that $f$ does not have an analytic continuation to any larger disk $D(0,r) , r > 1$
I'm not sure how to show the holomorphicity of a function like $f$ as it is a complicated infinite product. Are there any pertinent facts about infinite products to which I could appeal to? What about the second part about not having any analytic continuation?