It might be a silly question but I can't find how to prove this.
I assume this has a proof per axiomatic system but I am looking for proof which is most standard, using the most common axiomatic system.
e.g. when proving $\mathbb{R}_{+}$ is a group with multiplication, we usually just state it is closed without prooving.
To be more specific, how to prove it using ZFC?
Suppose the reals are defined as the metric completion of $\mathbb Q$, using Cauchy sequences. They inherit an ordering from $\mathbb Q$ and it suffices to prove that $\mathbb Q_+ \cdot \mathbb Q_+ \subset \mathbb Q_+$.
Now $\mathbb Q$ is defined as the fraction field of $\mathbb Z$. An explicit construction is as pairs $(a, b) \in \mathbb Z \times (\mathbb Z - \{0\})$ modulo the equivalence relation of being proportional by a nonzero integer. The multiplication is entrywise, and it is not hard to define $\mathbb Q_+$ once you have defined $\mathbb Z_+$, and to reduce the question to proving that $\mathbb Z_+ \cdot \mathbb Z_+ \subset \mathbb Z_+$.
Now $\mathbb Z$ is constructed out of $\mathbb N$ and (practically by definition) $\mathbb Z _+ = \mathbb N$, and $\mathbb Z_+ \cdot \mathbb Z_+ \subset \mathbb Z_+$ is then true because the multiplication on $\mathbb Z_+$ is the same as the one on $\mathbb N$.
