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For instance, if the sequence made up of all the second elements converge to some N, and all the third elements, and all the fourth elements to some N, etc, does the original sequence converge to N?

I considered using the fact that if subsequences converge then the original sequence converges but I don't know if I can apply that here because this is a nonstandard definition of subsequence.

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Define the sequence $x_n$ that is $1$ for prime indices and $0$ otherwise. Then $x_{2n}$, $x_{3n}$, $x_{4n}$, etc all converge to $0$. However, the sequence does not converge.

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    $\begingroup$ Haha you beat me by 6 seconds! $\endgroup$ Commented Nov 17, 2020 at 17:06

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