Question: 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 $b=a\circ p.$ 
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> 
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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: $p$.
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.