There is no number between 0 and 0.0*1, while both are not identical

BackgroundThe resolution of this debate states two things: (1) There is no number between 0 and 0.0*1, and (2) 0 and 0.0*1 are not identical.My opponent claims that I may prove the resolution wrong by naming a number between 0 and 0.0*1, however it is also sufficient that I prove 0 and 0.0*1 are identical. Unfortunately I will be unable to defend my argument from my opponent's rebuttals, whereas my opponent will be able to tailor his response to the claims I make. For this reason, previous statements made by my opponent on related topics will be considered indicative of the arguments my opponent intends to put forth, and the debate should be considered in its wider background.In his debate with Ajabi, my opponent claims that since the Real numbers are continuous, the existence of neighbouring Real numbers is “obvious.” Consequently, there must exist a difference between two adjacent Real numbers, and this difference must be positive, but infinitesimally small, or otherwise all numbers would be equal. Therefore, 0.9* and 1 are not identical; instead they are adjacent Real numbers and, according to my opponent, the difference between them is 0.0*1. [1]In this debate, I will argue the following: (1) the Real numbers are consistent and mathematically useful and should be used to judge this debate, (2) my opponent’s ‘neighbouring Real numbers’ are not a concept which can be derived from the axioms of the Real numbers and the allegedly obvious implication that the Real numbers contain non-zero infinitesimals is false, and (3) 0.0*1 and 0 are identical because there is no positive Real number between them. (1) The Real NumbersThe Real numbers are described by mathematicians as the complete ordered field. This is because the Real numbers are defined by the algebraic field axioms, the order axioms, and the continuity axiom, and it can be proven that any ordered field which satisfies these axioms must be isomorphic (i.e. they are the same field). [2]The consistency of the Real numbers is universally accepted by mathematicians. Gödel’s completeness theorem implies that a system is consistent if and only if it has a model (the model existence theorem), and the Real numbers can be formulated as a model, so therefore the Real numbers must be consistent. [3] The consistency of the Real numbers is so highly regarded that the Real number system is often used to prove the consistency of other mathematical systems (for example, Euclidean geometry) in proofs of relative consistency; in other words, if the Real numbers are consistent then so is a lot of other widely-used mathematics.Perhaps a simpler basis for accepting the consistency of the Real numbers is that inconsistency is death in mathematics. If the Real numbers were truly inconsistent or flawed, then any proof involving the Real numbers or relying on their consistency would be wrong. Consequently, if any mathematician could prove that the Real numbers were inconsistent then the Real numbers would be abandoned. The Real numbers are still widely used and no mathematician has contested their consistency, so it is reasonable to accept their consistency along with every other mathematician.Since my opponent specifically refers to the Real numbers in his debate with Ajabi, and uses them to form his basis for claiming that 0 and 0.0*1 are not identical but there is no Real number between them to conclude that 0.9* and 1 are not equal, the Real number system should be used to judge this debate. (2) Neighbouring NumbersMy opponent suggest that because the Real numbers are continuous, there must exist neighbouring numbers. As a result of the existence of neighbouring numbers, my opponent claims that there must also exist non-zero infinitesimals, and the difference between two adjacent Real numbers (for example 1 and 0.9*) is such an infinitesimal. [1]Curiously, the concept of neighbouring numbers actually makes more sense when considering discontinuous, incomplete number systems such as the natural numbers, {1, 2, 3,…}. For example, in the natural numbers, the number 2 neighbours the numbers 1 and 3, there is an indivisible, non-zero difference between them (the number 1.5 does not exist in the natural numbers). Clearly, neighbouring numbers is not a concept derived from continuity as my opponent claims, since it can easily be applied to discontinuous number systems. In fact, while neighbouring numbers is a concept which can be applied to discontinuous number systems, it cannot be applied to the Real number system because the Real number system is continuous.This is because the Real numbers are infinitely divisible. As a result of the continuity axiom of the Real numbers, between any two non-identical Real numbers, there are infinitely many other Real numbers. A simple proof by contradiction (see resources) that neighbouring Real numbers do not exist goes as follows: Suppose there exists two neighbouring Real numbers m and n such that the difference m-n = k>0 where k is a Real number. Then, following the field and order axioms of the Real numbers, there must exist a smaller difference, k/2. Hence there exists a number between m and n such that n < n+k/2 < m; however, this contradicts the assumption that m and n are neighbouring numbers. However, contrary to my opponent’s claim, this does not imply that all Real numbers are equal because the difference between them is still greater than 0; there is simply no smallest difference because there is no smallest positive Real number.My opponent attempts to get around this problem by proposing the existence of non-zero infinitesimals. However, the existence of non-zero infinitesimals contradicts the continuity axiom of the Real numbers. Using the continuity axiom, it is proven that the Real numbers are Archimedean which means they do not contain non-zero infinitesimals. [4][5] Zero is the only Real infinitesimal, however this poses a problem to my opponent’s argument: if neighbouring Real numbers exist and there is an infinitesimal difference between them, then the difference between any two Real numbers must be zero; alternatively if the difference between any two non-identical Real numbers is non-zero, then neighbouring Real numbers cannot exist.Consequently, my opponent’s concept of neighbouring Real numbers and Real, non-zero infinitesimals is false and self-defeating. (3) 0.0*1 and 0 are identicalThe notation 0.0*1 means that following the decimal point there are infinitely many zeros, and at the end of the zeros there is a one. This is a bit of a nonsensical notation, since if there are infinitely many zeros then there is no final zero onto which to append the one, but regardless this entails that 0.0*1 < 1/10k for every natural number k. In other words, 0.0*1 is an infinitesimal. As established above, in the Real numbers there are no non-zero infinitesimals; therefore 0.0*1 must be 0. The resolution is negated.References[1] http://www.debate.org...[2] http://www.math.umaine.edu...[3] http://tinyurl.com...[4] http://homepage.math.uiowa.edu... (pages 9-10)[5] http://www.math.vanderbilt.edu...Additional ResourcesA mathematically rigorous consideration of the consistency and completeness of RCF (which includes the Reals): http://www.personal.psu.edu...An introduction to the use of proof by contradiction in mathematics: http://zimmer.csufresno.edu...