A Variation of "rational is dense in $\Bbb R$" I know that the theorem that the set $\Bbb Q$ of rationals is dense in $\Bbb R$ says: 


  For every $x\in \Bbb R$ and every $\epsilon&gt;0$, there exist $a$, $b\in \Bbb Z$ with $b\ne0$ such that $$|x-{a\over b}|&lt;\epsilon.$$ 


But what about if I change the condition "$a,b\in \Bbb Z$ with $b\ne0$" by "$a$ and $b$ are both primes". Does the theorem still hold? In other words, I wonder whether the following is true: 


  For every $x\in \Bbb R^+$ and every $\epsilon&gt;0$, there exist $a$, $b\in \Bbb P$, where $\Bbb P$ is the prime numbers set, such that $$|x-{a\over b}|&lt;\epsilon.$$
Since $\frac{a}{b} &gt; 0$, yes.
Since $\frac{a}{b} &gt; 0$, yes.