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: 
Just to give a simple example with a single formula, let

$$a\_n={(1+(-1)^n)\sqrt2\over n}$$

The sequence is

$$0,\sqrt2,0,{\sqrt2\over2},0,{\sqrt2\over3},0,{\sqrt2\over4},\ldots$$

which clearly has both a rational and an irrational subsequence, each converging to the same limit, namely $0$.
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 $\sqrt 2$.