Definition of subsequence 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?

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).
