Sequences of Rationals and Irrationals 
*

*Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?

*Let $(y_n)$ be a sequences that converges to the rational number $y$. Must $y_1, y_2, \dots$ all be rational?
This was one of my midterm questions yesterday and I just wanted to clarify my responses. For (1), I said NO and as a counterexample, gave the sequence
$$
(x_n) = (3, 3.1, 3.14, 3.141, 3.1415, \dots)
$$
that converges to $\pi$ (note that each $x_j \in \mathbb{Q}$ since it is a finite decimal expansion). For (2), I said YES but was not sure how to prove it.
Could anyone verify these responses and if I'm correct about (2), offer a proof for why it must be true.

For (1), observe how $\mathbb{R}$ is usually constructed - as the set of all finite limit points of sequences in $\mathbb{Q}$. So not only is (1) wrong, you have that for every $x \in \mathbb{R}$ there is a sequence $(a_n)$ contained in $\mathbb{Q}$ such that $\lim_{n\to\infty}a_n = x$.
For (2) - if that was true, then only sequences of rational numbers could converge to zero. Yet, rather obviously, $\lim_{n\to\infty} \frac{\alpha}{n} = 0$ for every $\alpha \in \mathbb{R}$.
