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Let $(a_n)$ be a sequence of real numbers, and let $n_1<n_2< n_3 <n_4 <n_5 <···$ be an increasing sequence of natural numbers. Then the sequence $a_{n_1},a_{n_2},a_{n_3},a_{n_4} ,···$ is called a subsequence of $(a_n)$ and is denoted by $(a_{n_j} )$, where $j ∈\mathbb{N}$ indexes the subsequence.

What exactly is the meaning of "indexes the subsequence" ?

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  • $\begingroup$ By that j, you mean that you are only working with every j-th number instead all of them. $\endgroup$ – Atvin Mar 24 '15 at 19:00
  • $\begingroup$ Good question (can't upvote now, casted too many votes already). It means that they are good identifiers (in the programming languages sense of this word) for the subsequence. Indexes can be the natural numbers, the real numbers, ... . You have a series of values, and the indexes ("unique names") of the values are now given by natural numbers. For instance, sometimes you have a sequence with an uncountable number of elements. Then you use $j \in \mathbb{R}$ as indexes. You could even have used unique "words" to index them. It all doesn't matter. As long as all elements get a unique name. $\endgroup$ – Pedro Mar 24 '15 at 19:09
  • $\begingroup$ You use indexes to give unique IDs to things. In computer science this is something that is often done. You need to give a unique ID to every record in a database table. – Pedro 1 min ago edit $\endgroup$ – Pedro Mar 24 '15 at 19:09
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The original sequence is, technically, a function from $\Bbb Z^+$ to $\Bbb R$. Specifically, it’s the function that sends $n\in\Bbb Z^+$ to the term $a_n$ of the sequence; fall that function $a$, so that $a(n)=a_n$. To form the subsequence we have another function, which I’ll call $\varphi$, this time from $\Bbb Z^+$ to $\Bbb Z^+$; $\varphi$ is strictly increasing, and $\varphi(j)=n_j$ for each $j\in\Bbb Z^+$. Technically speaking, $\varphi$ is the sequence $\langle n_1,n_2,n_3,\ldots\rangle$.

In these terms the subsequence $\langle a_{n_1},a_{n_2},a_{n_3},\ldots\rangle$ of $a$ is just a composite function: it’s the composition $a\circ\varphi:\Bbb Z^+\to\Bbb R$, since for each $j\in\Bbb Z^+$ we have

$$(a\circ\varphi)(j)=a\big(\varphi(j)\big)=a(n_j)=a_{n_j}\;.$$

Saying that $j\in\Bbb Z^+$ indexes this subsequence is saying that we’re treating the subsequence as a function from $\Bbb Z^+$ to $\Bbb R$ and identifying each term of the subsequence by the $j\in\Bbb Z^+$ that is sent to it by the function $a\circ\varphi$.

Suppose instead that we let $M=\{n_j:j\in\Bbb Z^+\}$ and define a function $\psi:M\to\Bbb R$ by $\psi(m)=a_m$ for each $m\in M$. Each $m\in M$ is $a_j$ for some $j\in\Bbb Z^+$, so each $\psi(m)$ is one of the terms $a_{n_j}$. If we think of $M$ in its natural order as a set of positive integers, we can see that since $M=\{n_1,n_2,n_3,\ldots\}$, $\psi$ is the same subsequence $\langle a_{n_1},a_{n_2},a_{n_3},\ldots\rangle$ of $a$, but this time indexed directly by the set $M$. That is, this time we’ve the integers in $M$ directly instead of first indexing $M$ by the positive integers.

To take a concrete example, suppose that $n_j=j^2$ for each $j\in\Bbb Z^+$, so that $M=\{j^2:j\in\Bbb Z^+\}$. If we think of the subsequence as $\langle a_1,a_4,a_9,a_{16},\ldots\rangle$, we’re indexing it directly by $M$. If instead we think of it as $\langle a_{1^2},a_{2^2},a_{3^2},a_{4^2},\ldots\rangle$, we’re thinking of it as indexed by $\Bbb Z^+$, via the function $\varphi(j)=j^2$.

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