Edit: I really hope I'm not the only one laughing.
You are illustrating the reason that mathematicians generally disparage the concept of an "infinitesimal", when it's used as proof rather than conceptual aid. (Yes, I know about https://en.wikipedia.org/wiki/Non-standard_analysis)
Not only do you get wrong conclusions, you get tedious, hard-to-adjudicate arguments.
You mention that you know of non-standard analysis and indicate that it's irrelevant. Though I don't know why you think this, I agree with you. I just wanted to plug non-standard analysis as both mathematically interesting and also very useful. One can jettison the philosophical thought that NSA "vindicates" the historical use of infinitesimals (as I think we should), while still seeing NSA as the wonderful and deep piece of math that it in fact is.
And now every post I have made in this thread tree is getting downvoted. So I'm out. Y'all can argue about nothing--and nearly-nothing--by yourselves.
A number with infinitely many zeros after the decimal point is within 1/n of zero no matter the n. Therefore the number is zero.
This is like how 0.9999... is 1. The reason is that 1-0.999... is within 1/n of 0 no matter the n.
Suppose you had an infinitesimal epsilon (outside the real numbers --- this is fine, and people do this). How many times are you planning on adding it to itself? To get any actual real number, you are going to have to add it to itself well more than countably many times, though I'm not sure this makes much sense.