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You’ve been posting lots of vague and confused questions about sequences, derivatives, and cardinality to /r/math. You also have a habit of inventing our own terminology without motivating it or even acting as if folks should naturally understand it.
It seems clear from these threads (including this one) that you’re confused about some fundamental ideas surrounding sequences, cardinality, (un)countability, and differentiation. You’re getting so-so responses because folks have to nail down what exactly you’re asking/thinking before they can attempt to answer and that requires a ton of effort.
If you really want to learn this stuff, I’m of the mind that you need to spend less time posting ill-formed questions on /r/math and instead make sure you understand the fundamentals. For example, here some of the first things one typically learns when studying countability:
- The union of any finite number of countable sets is itself countable
- The union of a countable number of finite sets is itself countable
- The union of a countable number of countable sets is itself countable
Obviously (3) implies the first two, but each is progressively more difficult to prove for someone approaching these ideas for the first time. The latter two require some version of the axiom of countable choice, for example, which isn’t something most newcomers would think to deploy unless they had encountered it before.
They do, however, answer your question: if we have a countable set and “glue on” a countable number of countable collections of new numbers, the resulting set will still be countable.
I strongly recommend you buy and read Daniel Velleman’s How to Prove It. It will help you organize your thoughts better and help you get comfortable with the “standard” mathematical terminology and notation. Topic-wise it covers basic set theory and the last chapter is all about infinite sets, cardinality, (un)countability, and so on.
Here are some screenshots from Amazon’s “Search Inside the Book” to show you what you can expect by the end of the book: