I have been asked to show that for that

I know since and

Converge uniformly on compact subsets of the unit disk that both of the infinite products converge to a holomorphic function of the open disk.

So here is my idea let

and

I would be sufficient to show that

My idea was to take the deriative and show that it is equal to

let

This gives

I cannot sum this series or show that it is equal to 0. Also if you see a different approach please let me know.

Thanks