Is any subsequence of a sequence of rational numbers dense in $\mathbb{R}$

Question

We know that $\mathbb{Q}$ is countable so let $\{q_n\}_{n \in \mathbb{N}}$ a sequence of all rational numbers.

My Question is:

Is any subsequence $\{q_{n_k}\}_{k \in \mathbb{N}}$ dense in $\mathbb{R}$

Thanks.

Answer 1

"Any". Obviously. The sequence itself is a subsequence. And if you meant proper sequence just remove one term. Or take the sequence of those with even denominators, every neighborhood of a real contain a rational with an odd denominator.

"All". Obviously not. Take the $q_k$ that are integers.

Answer 2

This is equivalent to asking is any subset of $\mathbb Q$ dense?.

Some subsets are dense some are not e.g. $\mathbb N$