validity of proof $\pi$ is irrational from Euler's product formula

Question

Many articles and blog posts (and some textbooks) mention in passing that Euler's product formula can be used to prove that $\pi$ is irrational.

$$ \zeta(s) = \sum_n \frac{1}{n^s} = \prod_n \left( \frac{1}{1-p^{-s}} \right) $$

Taking $s=2$ gives us the Basel proble, and so:

$$ \frac{\pi^2}{6} =\prod_n \left( \frac{1}{1-p^{-2}} \right) $$

Since we know that there are infinite primes, and that each factor in the product is finite, we can say that $\pi$ can't be written as a product of finite rationals.

Is this correct?

This is the explanation given in several sources, but my concern is that we also need to show the infinite product can't be reduced, for example by "telescoping".

I would appreciate replies that were suitable for readers without university maths training.

Answer 1

No. Infinitely many positive rationals can have a positive rational product. For example, $\prod_{k=1}^\infty\frac{1-(k+2)^{-2}}{1-(k+1)^{-2}}$ telescopes to $\tfrac43$, which is an example of the concern you raise.