Is $\left\{ a \sqrt{3} - b \sqrt{2} | a, b \in \mathbb{N} \right\}$ dense in $\mathbb{R}$? I come along this problem while messing around with some other problems. So the question is:

Is $\left\{ a \sqrt{3} - b \sqrt{2} \ | \ a, b \in \mathbb{N} \right\}$ dense in $\mathbb{R}$?

I know for sure that:

*

*$A = \left\{ a \sqrt{3} + b \sqrt{2} \ | \ a, b \in \color{red}{\mathbb{Z}} \right\}$ is dense in $\mathbb{R}$.

*$B = \left\{ a \sqrt{3} + b \sqrt{2} \ | \ a, b \in \mathbb{N} \right\}$ is not dense in $\mathbb{R}$.

But what about $\left\{ a \sqrt{3} - b \sqrt{2} \ | \ a, b \in \mathbb{N} \right\}$? I suspect that it may be dense; however, I cannot prove it. Can someone please give me a push?
Thanks very much in advance, :*
 A: Note that if $\sqrt{2}a+\sqrt{3}b$ is dense in $\mathbb R$ then it is also dense if we disallow $(a,b)=(0,0)$.
Take $\varepsilon > 0$ and $c\in \mathbb R$. We must find an $a,b \in \mathbb Z$ so that $a\sqrt{3} + b\sqrt{2} \in (c-\varepsilon,c+\varepsilon)$ with $a$ positive and $b$ negative.
Take $a_1,b_1 \in \mathbb Z$ so that $a_1\sqrt{3} + b_1\sqrt{2} \in (c-\varepsilon/2, c+\varepsilon/2)$.
For each integer value of $a$ there is a minimum for $|a\sqrt{3}+b\sqrt{2}|$ where $b$ can take values on $\mathbb Z$, call it $f(a)$.
For each integer value of $b$ there is a minimum for $|a\sqrt{3}+b\sqrt{2}|$ where $a$ can take values on $\mathbb Z$, call it $g(b)$.
Let $\varepsilon'$ be a positive real smaller than $\varepsilon/2$ and smaller than $f(a)$ for all $|a| \leq |a_1|$ and smaller than $g(b)$ for all $|b| \leq |b_1|$, and smaller than $\sqrt{2}$.
Take $a_2,b_2\in \mathbb Z$ so that $a_2\sqrt{3} + b_2\sqrt{2} \in (-\epsilon',\epsilon')$, we can assume $a_2>a_1$ and $b_2< -b_1$ (otherwise multiply both by $-1$). Note that we must have $|a_2| > |a_1|$ because $\epsilon'$ is less than the $f(a)$ and we must have $|b_2| > |b_1|$ because $\epsilon'$ is less than the $g(b)$ and both values cant have the same sign because $\varepsilon' < \sqrt{2}$.
Notice $(a_1+a_2)\sqrt{3} + (b_1+b_2)\sqrt{2} \in (c-\varepsilon,c + \varepsilon)$ with $a_1+a_2$ positive and $b_1+b_2$ negative.
