# Need help with defining a subsequence

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)

• 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 $S.$ May 6, 2019 at 5:09

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.

• I don{t think this is what was asked... May 5, 2019 at 15:33
• @DonAntonio I’m a bit confused about what is being asked, then.. I’ll take this answer down if it is of no help. May 5, 2019 at 15:35
• 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. May 5, 2019 at 16:12
• So the indices are strictly increasing. So if sequence consists of an integer,then it must be constant since lim K=K?
– user669905
May 5, 2019 at 17:27
• 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
– user669905
May 5, 2019 at 22:37