Quadratic Integers over 2 form Subdomain of Reals/Proof 2

Proof
From Integers form Subdomain of Reals, $\left({\Z, +, \times}\right)$ is an integral subdomain of the real numbers $\R$.

We have that $\sqrt 2 \in \R$.

Every expression of the form:
 * $a_0 + a_1 \sqrt 2 + a_2 \left({\sqrt 2}\right)^2 + \cdots + a_n \left({\sqrt 2}\right)^n$

can be simplified to a number of the form $a + b \sqrt 2$, where $a, b \in \Z$.

The result follows from Set of Polynomials over Integral Domain is Subring.