Particular example of a non-vanishing analytic function f in unit disc |z|<1.

Question

This question is from Ponnusamy and Silvermann complex analysis section : Cauchy inequality.

Also please note that they are not homework problems. I am trying myself.

Question: Give an example of a non -vanishing analytic function f in unit disc |z| <1 with infinitely many zeroes.

Keeping uniqueness theorem in mind I need to find a sequence in |z|<1 with no limit point.

Now I choose $ a_1 =0.1 , a_2=0.11, a_3 =0.111, a_4 =0.1111$ and so on.

I think this will not have any limit point but that's due to the reason that I am unable to find limit point of this sequence (I think it doesn't exists).

But can you please prove rigorously why limit point doesn't exists?

If I am wrong kindly tell an example that will work.

Many thanks!!

Answer 1

You are searching for $$\prod_{n=1}^\infty (1-\frac{10^{-n}}{1-z})$$ Does it converge, is it analytic, where are its zeros