I'm having a little difficulty wrapping my head around the difference between these two terms, so could someone verify if this is correct? I've struggle to find an answers or reference in a book or online.

Component sequence:

If you have a set $X \subset \mathbb{R}^m$ and a sequence $x \in X^\infty$ such that

$$x = (x\_1, x\_2, \dots ) \in X^\infty = (\mathbb{R}^m)^\infty$$

Each $x\_i$ is itself a vector of $m$-terms, 

e.g., $x\_1 = (x\_{1,1}, x\_{1,2}, ... x\_{1,m})$, or for another notation: $x\_1 = (x\_1(1), x\_2(1), ... x\_m(1))$ 

i.e., each term of the sequence is itself an m-tuple. Then, a component sequence is defined as the $k-th$ component of each term $x(n)$, and hence, there are $m$ of these component sequences, a sequence for each component.

But what does that actually mean, practically? If I have a sequence:

$$ S = (x\_1, x\_2, x\_3 , ...)$$
where $x\_1 = (1,3,5,7,9)$, is one such component sequence of $S$ just $(1,3,5,7,9)$?

As opposed to a subsequence which could be $(x\_1, x\_3, x\_5, ...) $ ?
['https://math.stackexchange.com/questions/3326203', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/466286/']
Component sequences are projections of your sequences on the one dimensional axes of your space. Each component sequence is an infinite sequence of picking the sequence of terms in one dimension of the original sequence . 
For example if $$\{(1,4,5),(3,2,4),(5,3,6),...\}$$ is a sequence in $R^3$, then you have three component sequences namely $$\{1,3,5,....\}, \{4,2,3,....\},\{5,4,6,...\}$$
The answer to you example-question is no. A component-sequence takes only one component per term, always from the same position, and so for each term.

So, if $x\_1 = (1,3,5,7,9)$ like you suggest, then there are exactly five component sequences of $x$, one of them starts with $1$ and takes the first coordinate of each term, another starts with $3$ and takes the second coordinate of each term, etc.