# Why does it suffice to show it for positive integers?

I am looking at the proof of the product formula theorem:

For each $x \in \mathbb{Q}$, it holds $$\prod_{p \leq \infty} |x|_p=1$$

The proof starts by this:

It is enough to show it for each positive integer $x$.

Could you explain to me why when we show it for each positive integer $x$, then we have shown that it holds for each $x \in \mathbb{Q}$ ?

Because the norm is multiplicative: $$\prod_p \left |\frac x y \right |_p = \left(\prod_p |x|_p\right) / \left(\prod_p |y|_p\right)$$
Just make sure that you understand why this formula is true for $x= -1$!