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?

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. 
