Question: The original question: If we have a sequence of real numbers and on that, we define a sequence of $n^{th}$ rational numbers and another of irrational numbers, then they are both subsequences of the original sequence of real numbers?

In other words, can a rational sequence be a complementary subsequence of an irrational sequence?

That is can there exist a sequence of real numbers which has both a rational subsequence and an irrational subsequence?

Answer: 
Consider $(a\_n)$ given by

$a\_{2n}= \frac{\sqrt{2}}{2n}$ and $a\_{2n-1}= \frac{1}{2n-1}.$
Consider the sequence $$1, 1.4, 1.41, 1.414, 1.4142\ldots$$ that converges to $\sqrt 2$, and the sequence $$\sqrt 2,\sqrt 2,\sqrt 2,\sqrt 2\ldots$$ that also converges to $\sqrt 2$.