Question: I need help understanding the following definition:

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> 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 $b=a\circ p.$ 
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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?

Answer: 
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: $p$.
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).