I have seen in various posts both
0,9999 and
0,9999...(meaning that we are imaginig infinite 9's)

Well, here is the question.
What do you mean by saying that a given number is given by an infinity of digits? In other words, which is the meaning of the decimal representation of a number, when it is not finite?
The number is the limit of the partial ways of writing. In our example, to say that 0,9999999........ = 1
means exactly that 1 is the limit of the succession
0,9
0,99
0,999
0,9999
........and so on