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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 $\lim_{n \to \infty} x_n \ne \lim_{n \to \infty} y_n$;
  1. $x_n$ 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 $(2)$?

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  • $\begingroup$ A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,... $\endgroup$ – Mark S. Nov 25 '18 at 14:02
  • $\begingroup$ What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R? $\endgroup$ – Paul Nov 25 '18 at 14:03
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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 $$ (y_n)_{n=1}^\infty = (1,0,1,0,\ldots). $$

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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 $y_n = (-1)^n$

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