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 ::I do not need to supply any references to show that your definition of infinitesimal is false. All I have to do is set n = infinity and already I have problems. As far as infinitesimals belonging or not belonging to the reals, it is you who have to take your pick! Firstly, you use the plural form - this is a contradiction in itself. Are there infinitesimals that are smaller than other infinitesimals? Secondly, the completeness principle states that every non-empty set which is bounded from above has a LUB. If you are using this, which you are, then you have to make up your mind whether you are using infinitesimals or not. I am not ignorant of mathematics so your discussion about fields above reveals nothing that I did not already know. Had you read my response in context (which you did not), you would have understood that I am claiming pi, e and sqrt(2) are part of the reals and these numbers have something in common with 0.999..., 0.333... in that they can only be appromixated. You archived the previous posts in which I provided *valid* mathematical proof (nothing hand-waving about this). Go back and read the posts. I pointed out errors in your 'proof' and when you realized that you had in fact written rubbish, you resorted to Wiki's NOR policy. An easy way out for you? 192.67.48.22 

 :::Sigh. Obviously n has to be a natural number. But even if we assigned some meaning to n = infinity, the proof would still work. There are no real x&neq;0, so that . As there are no infinitesimals in the reals, we cannot really discuss their properties (in a field that contained infinitesimals, however, it would be trivial to prove that there would be at least countable many). You still haven't given me your definition of infinitesimals or of the Archimedean property. You haven't shown how the Archimedean property implies the LUB property. It looks like you didn't really mean that ""If the infinitesimal does not belong to the reals, then pi, e or any other number with similar properties does not belong to the reals also"", though. There are no formal mathematical proofs in the archive. Feel free to make one here, however. WP:NOR just means that even if you should be able to produce a valid proof for 0.999... &neq; 1 and convince everybody here that it is correct, you do not get to move the article to Proof that 0.999... does not equal 1. You would have to produce a reputable source. It doesn't mean that we can't produce lots of talk-page material, that we can move to another archive, once we are done.   "