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 can not 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