In the above example, .99999 was carried only to n=50 th decimal point, whereas n=1,000 was used in the narrative. How close does this come to what has been described as ‘hyperreal’, depends on the definition of what is considered to be ‘real’.
The same kind of analysis can be used between limitless and limited functions, it is limitless until a limit is used. The difference between the two types of propositions, is the first is a quantifiable and the second is a qualifiable one.
The former uses numerical functional predecessors of at least .1, whereas the later uses the presumption of the sum of all partial differentiations=1 .
If presumption for the second is to hold water, then that set is inconclusive as to whether the sum can ever be assumed to be 1, because an infinity of .9’s has to be assumed. The other kind of infinity only proposes, that in the function of n sequences of 9’s, where n~infinity, there can not be any closing 0 number.
Therefore the proposition that n number of sequences of .9’s where n=1~infinity, is only a linear real set, whereas the set described as a sum of all members of the set, where the same proposition leads to the sum of all the differentiated members of the set are not linear but differential.
The two types of sets are not differentiable by the way their functions are defined. The former function only points to linear succession , of a simple linear sequence of the type- x~x~~…whereas the second involves the type- : the sum of the above =x+x+x…+x=1.
Hypothetically, a linear sequence, proposing nothing else then the ongoing sequential progression of a addition of a number which diminishes value by a set constant rate, does not imply a different(ial) function
that has a limit.
If that different function could be inferred, that would require the function limit the function into a real limit using real numbers. The problem with the difference between real and hyper real numbers are, that they can not be said to differ except , how their function , or use be relegated.
It is like saying, any differential value has a real based set, until that function becomes useless. But since derivatives can only be derived backwards, in terms of differentiating the useful from the useless, until that time to talk of useful functions makes no sens. But that is all that differentiation can do, it can not work forewards. Until that time, all math is merely intuitive.