Unable to understand proof for Infinitude of primes

Question

I have got a proof for showing the infinitude of prime numbers. I am facing issue where the book states that: "Thus, $p_i$ must divide $1$, so then $p_i=1,$"
I feel there is no reason given to understand why $(s-t)$ cannot divide $1$.
I state that for equality of the product of $(s-t)p_i$ to $1$, both $s-t=1$, & $p_i=1$, as all $s, t, p_i$ are naturals. The book has shown failure for $p_i$, with no proof given for failure of $(s-t) \ne 1$.
If $s-t=1$, then $s = p_1p_2\cdots p_{i-1}p_{i+1} \cdots p_n +1$.
So, my analysis stops at: it reduces to stating that $s \ne p_1p_2\cdots p_{i-1}p_{i+1} \cdots p_n +1$.

Answer 1

The book states $1=(s-t)p_i$.

Therefore $s-t$ can, and does, divide $1$. But that is not at all the point.

Answer 2

Since $(s-t)p_i=1$, since $s-t\in\mathbb Z$ and since $p_i\in\mathbb N$, $s-t=p_i=1$.