A subset $S$ of $\mathbb{R}$ is said to be dense in $\mathbb{R}$ if $\forall$ $\epsilon > 0$ and $x \in \mathbb{R}$, $\exists$ $s \in S$ such that $| x - s| < \epsilon$.
Using this definition, I would like to try to prove that the set $S = \{ a + b \sqrt{3} : a ,b \in \mathbb{Z} \}$ is dense in $\mathbb{R}$.
Any help would be appreciated.
Quick way to show that $S$ is dense in $\Bbb{R}$:
$S = \{ a + b \sqrt{3} : a ,b \in \mathbb{Z} \}$ contains the element $\zeta := 2 - \sqrt{3}$
Since $S$ is closed under addition and multiplication, $\zeta^n$ is in $S$ for every $n \geq 2$
For any $x \in \Bbb{R}$, $x$ is at most $\zeta^n/2$ away from some element of $\zeta^n \Bbb{Z} \subset S$
Since $0 < \zeta < 1$, $\zeta^n \to 0$ as $n \to \infty$.
HINT: It follows from the fact that $\{nr\bmod 1:n\in\Bbb Z\}$ is dense in $[0,1)$ whenever $r$ is irrational. (Here $x\bmod 1=x-\lfloor x\rfloor$ is the fractional part of $x$.) This fact can be proved using the pigeonhole principle; I gave a proof here.
