Let me try to explain why (0.333\dotso) is not an infinite sequence.
When we ask “Is (0.333\dotso) an infinite sequence?” what we’re asking is “Does the symbol (0.333\dotso) represent an infinite sequence?”
This means that we’re asking what the symbol (0.333\dotso) represents and NOT what the symbol (0.333\dotso) is in itself.
When we ask “Are numbers sequences?” what we’re asking is “Does the word represent a sequence?”
Since the word “number” does not represent itself, we do not care about the fact that the word “number” is a finite sequence of letters. In other words, the fact that the word “number” is a finite sequence of letters does not mean that what the word “number” represents (= symbolizes = signifies) is a finite sequence of letters (or a sequence at all.)
In the same exact way, the fact that the symbol (0.333\dotso) is a finite sequence of characters does not mean that it represents a finite sequence of characters (or a sequence at all.)
That symbol, on its own, is a sequence consisting of (8) characters. But the symbol itself does not represent a sequence of (8) characters.
(0.333\dotso) is a symbol representing certain number. And it’s not the only symbol that represents that number. There are many other symbols representing the same number. There are finite sequences of characters such as (0.3 + 0.03 + 0.003 + \cdots) as well as infinite sequences of characters that we cannot write down (since they are infinite) but that we can represent using other symbols (e.g. what is represented by the statement “a sequence that starts with (0) followed by (.) and an infinite sequence of (3)'s” is an example of such a symbol.)
Anything can be represented using any kind of symbol. The fact that you can represent a number using an infinite sequence DOES NOT MEAN that that number is an infinite sequence.
You can represent numbers with horses. That does not mean that numbers are horses.
The symbol is not the symbolized.