Determining $\lim\sup_{n→∞} q_n$ and $\lim\inf_{n→∞}q_n$ for a countable set $\mathbb Q ∩ [0, 1]$

Question

Could anyone help me with this question?

Let $\{q_n\}_{n∈\mathbb N}$ be an enumeration of the countable set $\mathbb Q ∩ [0, 1]$. Determine $\lim\sup_{n→∞} q_n$ and $\lim\inf_{n→∞}q_n$

So my guess is that any element in this interval is an accumulation point, so $\lim\sup_{n→∞} q_n= 1$ and $\lim\inf_{n→∞}q_n = 0$. But how can I prove that?

Answer 1

according to the definition of an accumulation point, you would need to proof that every element of {q} is a partial sequence that converges, so that every element of {q} is an accumulation point. but how to show that? this follows somehow out of the statement that Q is dense in [0,1], does it?