That's completely correct, but likely to not be useful. It's also not uncommon, but ... only during the initial formalizing of sequences, after which we put it aside. Folks seldom think about things like cartesian products of sequences, because they don't correspond to "natural" operations on sequences. (An example of a natural operation on two real-number sequences might be "termwise addition"; another might be "interleaving.")
If I said "Your cellphone is entirely made up of atoms," it'd be true, but it wouldn't give you a clue about why the atoms were arranged in the way they were by the designer of the phone.
In much the same way, it's nice to find that something we regard as natural ("a sequence of stuff, one thing coming after another") can be represented by the things we've agreed to use as the basis for our mathematics (sets), but as soon as you realize that they can be represented this way, it's useful to go back to thinking about them the way you always have, rather than trying to always understand them in the most basic form.
In much the same way, everything in a computer is represented by 0s and 1s, but it'd be a nightmare to try to think about how the particular pixel on your screen that forms the upper left corner of the first digit of my phone number corresponds to a particular +3.3V on some wire at some instance. The joy of abstraction is that once you realize that using the voltages 0v and 3.3V (or some other standard), you can encode the values 0 and 1, and that you can build circuits that, say, compare these values, or add them, or subtract them, etc., you can stop thinking about electrons and think only about 0s and 1. Soon after that, you can think about 8-bit sequences of 0s and 1s (called bytes), and then think about particular ways of organizing these to represent certain not-too-large integers, or characters in a font, etc.
The one exception I've personally found is that when I'm given a challenging probability problem, I find it useful to see whether I can make it actually fit into the precise definitions I know for things like random variables (a function from a measure space to the real numbers (or small generalizations of this)).