yes i am saying that this follows from the previous steps in complete agreement with previous assumptions i will not discuss the infinitesimal because it is nonsense there are no infinitesimals in the real numbers or any other numbers non standard numbers are also nonsense i have illustrated clearly what happens when you treat a 'false' assumption as true you end up with a contradiction the false assumption here is that it is not possible to find an epsilon just as it is impossible for you to completely represent an irrational number you don't seem to have a problem with using pi or other irrational numbers as approximations but you have the audacity or rather foolishness to make wild claims like the ones in your main article by definition there is no duplicate representation it is your attempt to have this equality that gives rise to duplicate representation quite frankly it does not matter that we can't find an epsilon because our arithemetic deals only with finitely represented quantities or approximations in all other calculations where we cannot completely represent given numbers so which proof do you want me to illustrate as false next your 'archimedean type' proof