Necessary or sufficient conditions for rationality of a limit of a sequence of rational numbers Consider a convergent sequence of rational numbers $a_n$ with a limit $\lim_{n\rightarrow\infty} a_n = b$.
Does there exists some kind of necessary condition for $b$ to be rational that only uses the elements of the sequence?
It is easy to describe sufficient condition for integers, like $\forall n\in \mathbb N, \ a_n \in \mathbb Z \implies b \in \mathbb Z.$
But this is definitely not true for rationals: $\forall n\in \mathbb N, \ a_n \in \mathbb Q \not\implies b \in \mathbb Q.$  
Also it is true that: $\forall n\in \mathbb N, \ a_n \in \mathbb R \implies b \in \mathbb R.$
Are there sufficient conditions for $b$ to be rational that are not of the form $$\exists k \in \mathbb N: |\{ n \in \mathbb N \ |  \ a_n = a_k \}| = \infty?$$ 

It is not true that: $\forall n\in \mathbb N, \ a_n \in \mathbb R \implies b \in \mathbb R.$ (I assume that by $x\in \mathbb R$ you mean $x\in \mathbb R - \mathbb Q$)
For example $\ a_n = 1/n \pi$ converges to $0 \in \mathbb Q$ and $\ a_n = p/q + 1/n \pi$ converges to $p/q \in \mathbb Q$ for any p/q.
I am thinking that looking at the extended fraction representation might help in your initial question, as rational numbers the fraction terminates but for irrational numbers it is infinite.
