A problem about a sequence of sequences whose limits is a convergent sequence Let $(x^r)_{r \in \mathbb{N}}$ be a sequence such that $x^r \to x^*$:
$$x^1\quad x^2 \quad...\quad x^r \quad...\quad \to  \quad x^*\quad \quad(r \to \infty)$$
Suppose thar for ecah $r$, there exists a sequence $(y^r_n)_{n \in \mathbb{N}}$ such that $y^r_n \to x^r$, as $n \to \infty$.
I want to show that there exist a subsequence $(n_r)_{r \in \mathbb{N}} \subset \mathbb{N}$, such that:
$$ y_{n_r}^r \to x^* $$
Thanks

For every $r$, since $y_n^r\to x^r$, there exists an $n_0(r)\in\mathbb N$, such that
$$
n \ge n_0(r)\quad\Longrightarrow\quad |y_n^r-x^r|<\frac{1}{r}.
$$
Now $z^r=y_{n_0(r)}^r\to x^*$, since
$$
|y_{n_0(r)}^r- x^*|\le |y_{n_0(r)}^r-x^r|+|x^r- x^*|,
$$
and
$$
|y_{n_0(r)}^r-x^r|\to 0\quad\text{and}\quad
|x^r- x^*|\to 0
$$
as $r\to\infty$.
