Example of sequences $\{x_n\}$ and $\{y_n\}$ with matching ranges of values but different limits.

Question

Find an example of two sequences $\{x_n\}$ and $\{y_n\}$ such that $R(x_n) = R(y_n)$ and:

1. $x_n$ and $y_n$ are convergent, and $\lim_{n \to \infty} x_n \ne \lim_{n \to \infty} y_n$;

2. $x_n$ converges and $y_n$ diverges.

Here $R(x)$ denotes the set of all possible values of the sequence. Roughly speaking $y_n$ is a permutation of $x_n$ (at least as far as I understood it).

For the first case I was thinking of such a function which would have the same range. For example consider the following sequences:

$$ f(x) = \arcsin\left(\frac{x-2}{x-1}\right) \\ y(x) = \arcsin\left(-\frac{x-2}{x-1}\right) $$

Clearly both of them have the same range of values but their limits are different. The problem is that replacing $x$ with $n$ breaks continuity and we get this.

Second case is unclear to me.

Could we find some sequences satisfying $(1)$ and $(2)$?

Answer 1

No need to consider such complicated functions. For (1), just take $$ (x_n)_{n=1}^\infty = (1,0,0, 0,\ldots), \quad (y_n)_{n=1}^\infty = (0,1,1,1,\ldots), $$ and for (2), keep the same $(x_n)$ and take $$ (y_n)_{n=1}^\infty = (1,0,1,0,\ldots). $$

Answer 2

Example for Case 1:
Let $x_0 = -1,\ x_n=1$ for $n \geq 1$
Let $y_0 = 1,\ y_n=-1$ for $n \geq 1$

Example for Case 2:
Let $x_0 = -1,\ x_n=1$ for $n \geq 1$
Let $y_n = (-1)^n$