Two divergent sequences like $\{x_k\},\{y_k\}$ such that $\lim_{k \rightarrow \infty}|y_k-x_k|=0$

Question

A sequence with a limit that is a real number is called a convergent sequence.

A sequence which does not converge is said to be divergent.

Find two divergent sequences like $\{x_k\},\{y_k\}$ such that $\lim_{k \rightarrow \infty}|y_k-x_k|=0$ .

Notice that we know $\forall k \in \mathbb N \space 0 \lt |y_k-x_k|$

Note ( For those who ask about my try ) : There is nothing to try! If my try was successful, I wouldn't be asking this question.

Answer 1

$$ x_n=(-1)^n+\frac1n\quad\text{and}\quad y_n=(-1)^n. $$

Answer 2

It's not clear to me why there is "nothing to try"; you can certainly try to think of two divergent sequences that have a minor (or no) difference.
For example, take $x_k=y_k=k$, or less trivially, $x_k=k$ and $y_k=k+\frac1k$