In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$.
Why is that enough to prove that it is dense on $\mathbb{R}$ ?
In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$.
Why is that enough to prove that it is dense on $\mathbb{R}$ ?
Because then you can take the integer multiples $k(a+b\sqrt2)=ka+kb\sqrt2$ to fill the rest of the line.