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How would you define the definition of a subsequence of $x_n$ if $x_n$ is an integer?

By definition it increasing subset of natural numbers from $x_n$ For instance let $x_n=(c,c,c..)$ Any subset of this sequence the elements are constant when $k_1$<…<$k_n$

So I was thinking if the elements are constant,does that infer that the indices should be equal That is what I was trying to say.

(I hope I got my point through)

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  • $\begingroup$ Think of a sequence as a function $f$ with domain $S$ where $S$ is an infinite subset of $ \Bbb N$ (or of $\Bbb N\cup \{0\}$) and a sub-sequence of $f$ as the restriction of $f $ to a domain $T$ where $T$ is an infinite subset of $S.$ $\endgroup$ May 6, 2019 at 5:09

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Let $(x_n)$ be a sequence. Then a subsequence of $(x_n)$ is a sequence $(x_{k_n})$, where $k_n$ is a strictly increasing sequence of natural numbers.

I think you’re getting confused between the index of the sequence and the value of the sequence itself. The definition you have been given is saying the equivalent of what I have said above—that $k_1 < k_2 < \dots$ or that the sequence of indices to the sequence is strictly increasing. It doesn’t say that the value of the sequence is strictly increasing.

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  • $\begingroup$ I don{t think this is what was asked... $\endgroup$
    – DonAntonio
    May 5, 2019 at 15:33
  • $\begingroup$ @DonAntonio I’m a bit confused about what is being asked, then.. I’ll take this answer down if it is of no help. $\endgroup$ May 5, 2019 at 15:35
  • $\begingroup$ Well, trying to make some sense from what is written in the question, I think the OP is trying to ask the following: if $\;\{x_n\}\subset\Bbb Z\;$ is a sequence s.t. $\;\lim\limits_{n\to\infty} x_n=c\in\Bbb R\;$ , then there exists $\;M\in\Bbb N\;$ such that for any $\;n\ge M\;$ the sequence $\;\{x_n\}\;$ is constant (and, in fact, equal to the limit $\;c\;$ ) . The OP messes up with subsequences as the question may have been given that way, but it is just the same, since any subsequence of a converging sequence converges and\ to the same limit. $\endgroup$
    – DonAntonio
    May 5, 2019 at 16:12
  • $\begingroup$ So the indices are strictly increasing. So if sequence consists of an integer,then it must be constant since lim K=K? $\endgroup$
    – user669905
    May 5, 2019 at 17:27
  • $\begingroup$ Thanks I guess I got confused. DonAntonio got it right as the first person. I thought the values related to the indices and thier ordering $\endgroup$
    – user669905
    May 5, 2019 at 22:37

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