Product Formula for Norms on Non-zero Rationals

Theorem
Let $\Q_{\ne 0}$ be the set of non-zero rational numbers.

Let $\Bbb P$ denote the set of prime numbers.

Let $a \in \Q_{\ne 0}$.

Then the following infinite product converges:
 * $\size a \times \displaystyle \prod_{p \mathop \in \Bbb P}^{} \norm a_p = 1$

where:
 * $\size {\,\cdot\,}$ is the absolute value on $\Q$
 * $\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q$ for prime number $p$

Lemma
Let $a = \dfrac b c$, where $b, c \in \Z_{\ne 0}$.

From the Lemma, the following infinite products converge:
 * $\size b \displaystyle \times \prod_{p \mathop \in \Bbb P} \norm b_p = 1$
 * $\size c \displaystyle \times \prod_{p \mathop \in \Bbb P} \norm c_p = 1$

From Quotient Rule for Real Sequences, the following infinite product converges:
 * $\displaystyle \dfrac {\size b} {\size c} \times \prod_{p \mathop \in \Bbb P } \dfrac {\norm b_p}{\norm c_p} = \dfrac 1 1 = 1 $

We have: