Will an infinite union of rational numbers will the real line?

Question

Consider $\mathbb{R}$ as the underlying metric space with the usual metric. Suppose we construct sets:

$I_1 = \mathbb{Z}$

$\displaystyle I_2 = \{\ldots, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, \ldots\}$

$\displaystyle I_3 = \{\ldots, -1, -\frac{2}{3}, -\frac{1}{3}, 0, \frac{1}{3}, \frac{2}{3}, 1, \ldots\}$

$\ldots$

Now let $I = \cup_{n = 1}^{\infty} I_i$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, does $I = \mathbb{R}$?

Answer 1

Note that $I$ is just $\mathbb{Q}$. So you're simply asking whether there is an irrational number. Or if you prefer:

Where does, say, $\sqrt{2}$ occur in your $I_n$s?

More generally, "$A$ is dense in $B$" in no way implies, or even should suggest, that $A=B$.