Can we let $n$ go to $\infty$ after applying the fundamental theorem of algebra to a polynomial of degree $n$?

Question

Recently I came across a proof of the infinite product for $\sin z$ (https://www.sciencedirect.com/science/article/pii/0022247X77902347). It applies the fundamental theorem of algebra to $$p_{n}(z)=\dfrac{1}{2}\left(\left(1+\frac{z}{n}\right)^{n}-\left(1-\frac{z}{n}\right)^{n}\right),$$ which it factors to a product. But it also states that $n=2m+1$ and lets $n$ and $m$ go to infinity.

Then how is the factorization to a product, by applying the fundamental theorem of algebra, possible?

I'm confused because the fundamental theorem of algebra holds only for finite polynomials, but $p_{n}(z)$ is not a finite polynomial if we let $n$ go to $\infty$.

Answer 1

You are of course right that

the fundamental theorem of algebra holds only for finite polynomials,

I suspect that if you read that paper carefully you would find that Eberlein's purpose is (in part) to prove rigorously that Euler's seemingly casual "letting $n$ go to infinity" produces mathematics that's correct by 20th century standards.