But you cannot do that here whiteflags. It is exactly this closeness between an infinitesimal and 0, or 0.999... and 1 that we are discussing. It may be true for certain types of operations where limits are used to facilitate calculations. But it is wrong to propose a proof that 0.999... is 1 by using a method in which we simplify 0.999... to 1. Or an infinitesimal to 0. What use can that be to anyone?

Here is something a lot more interesting. Why, I think, a certain level of rigor and a more critic attitude to seemingly established concepts serves us better:

Is 0.999... = 1?

I don't propose to understand all that is being discussed. But clearly here the author understands that under certain scenarios 0.999... may be equal to 1, if we allow ourselves to introduce a margin of error into our calculations. But as an universal truth, we cannot. We are then forced to introduce other concepts such as, and I quote:

The acceptance that we musty introduce new concepts such as 0¯ or 1¯ and that our mathematical operations are powerless to deal with these quantities (real numbers) are fundamental to accept certain proofs into our lives without that meaning we should defend them as universal in the domain of Real numbers.Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real.

In other words: A little more humility serves better anyone pretending they can prove 0.999... is equal to 1.