Define $$\epsilon_Q = 1/3 - 0.333... \\ \epsilon_c = 1 - 0.999... $$ in honor of QQ and the OP.

Then it follows that $$3 \epsilon_Q = 3 \times ( 1/3 - 0.333... ) = 1 - 0.999... = \epsilon_c$$ so $$3 \epsilon_Q = \epsilon_c$$, right?

So why is $$\epsilon_Q \neq \epsilon_c$$?

Doesn't your scheme require an infinite number of infinitesimals if $$\epsilon_c \neq 0$$ ? How is this an improvement on the Real Numbers which have no such infinitesimals?

I am not sure that I understand the question fully, [perhaps one day I will understand the notation you are using] but I tend to believe that the use of an infinitesimal is a way that mathematics can resolve the paradox using the legitimate "fudge" factor of 1(1/infinity).

It has to do I think with the philosophical question raised by Zeno's paradox and acutely demonstrated in the notion of 10/3 or 1/3 using base 10.

*Extended:*
Philosophically it sums up to the ability for mathematics to prove that "zero" is indeed a null value.

From what I have come to understand after exhaustive philosophical discussion, is that zero can only be proved as a "nul" quantity or value with the use of infinitesimals as being the smallest quantity or value, we can go with out arriving at zero. Thus proving zero only by default and not directly. Deductive reasoning or logic? perhaps...

An infinitesimal is unable to be rationally multiplied or divided as it's definition is always going to be 1(1/infinity) regardless of the number of times it is used.

Same argument would I think apply to infinity * infinity = infinity...

To cut a portion of cake exactly 1/3 [in absolutum] is impossible [just like it is to resolve Pi] as infinite reduction towards the infinitesimal, as a part of measuring that 1/3, is all that is possible.

The other issue worth mentioning is that looking at any whole number... it may be realised that the "wholeness" is only available when used in the context of "Quantity" and not "Quality" [aka value]

Example:

We have 10 apples each weighing 1 kg.

10 apples is true [quantity]

1 kg exactly is false [ due to infinite reduction.] [quality - value] Suffice to say it can never be exactly 1kg [in absolutum]

The issue displayed in the process Sarkus mentioned is I believe erroneous due to the fact that it mixes quantity with quality and there is a subtle but important loss of contextual consistency in that process. [as exampled above with the apples]