I need help understanding the following definition:
We say that a sequence $b:\Bbb N\rightarrow S$ is a subsequence of a sequence $a:\Bbb N\rightarrow S$ if there exists a strictly increasing sequence $p:\Bbb N\rightarrow \Bbb N$ such that $b=a\circ p.$
So if I take for example that $a$ is a sequence of natural numbers and $b$ is a sequence of even numbers, then what is $p$ here?
In this case, $p$ is exactly like $b$:
$$b = {2,4,6,8,...}$$ and
$$a = {1,2,3,4,5,6,7,8,...}$$
then $p$ would denote the indices to select from $a$ in order to create $b$
the even numbers are at the second,fourth,sixth,... indices: ${1,(2),3,(4),...}$ where the numbers in $()$ are the even indices.
then the sequence $b$ is a sequence of selected values, and the way they are selected is related to the indices, which are presented as a sequence of integers: $p$.
Just take a $k\in \Bbb N$ and send it to $2k$, in symbols $p(k) =2k$
If $b$ is the even naturals, and $a$ is the naturals, then $$ p:\mathbb{N}\rightarrow \mathbb{N}\\ k\mapsto2k $$ and thus $a\circ p:\mathbb{N}\rightarrow\mathbb{N}=S$ is exactly the even terms in the sequence $a$. The increasing condition is to make sure what you get is really a "sequence" in the sense that you have some indexing set that marches along.
Assuming that b is indeed a subsequence of a, then p is the list of the indices of the members of a, that are in b. This is just the general definition.
However I assume that you had a slightly different example in mind, namely that b is the subsequence of a, of all the even numbers of a. p is therefore the list of the indices of the members of a, that are even.
Example: a = 1,3,2,4,9,2,...., and therefore b = 2,4,2, ...., and therefore p = 3,4,6,... In this example a does not follow an arithmetic pattern.
(I apologize for writing in plain. I don't know how to format properly).