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.

Both should be no. For (2), just divide by $\pi$ of your first example.
