Let $A=\{a+b\sqrt{2}\mid a,b\in\mathbb Z\}$ and let $Y\subset A$ with $x\in Y$ iff $x\in [0,1]$.

I'm trying to prove that $A$ is dense in $\mathbb R$ and already noticed that it is enough to show that $Y$ is dense in $[0,1]$. Could any one point me in a direction?

I have already looked at a proof of the density of $\mathbb Q$ in $\mathbb R$, where the core of the proof consisted of showing that between any two reals there is a rational. This was done by multiplying the reals with a natural number until the difference between them became bigger than one and noticing that you can find a rational between the two multiplied reals and divide that by the amount the reals got multiplied with and you get a rational between the two reals.

This method does not work identically in this case, because dividing an element in $A$ by a natural number won't always yield another element in $A$.

I will use this, when I have a proof, to prove that the quotient $\mathbb R/{\sim}$ with $r\sim s$ if $r-s=a+b\sqrt{2},\;\exists a,b \in \mathbb Z$ is not hausdorff. The density of every eq. clas in this quotient implies that the quotient topology must be the trivial one, hence the quotient cannot be hausdorff.