and the first equality established above, simplifies to This implies that the difference between and is less than the inverse of any positive integer. Thus this difference must be zero, and, thus ; that is This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all . (This is implied by the fact that implies ). Algebraic arguments Many algebraic arguments have been provided, which suggest that They are not mathematical proofs since they are typically based on the fact that the rules for adding and multiplying finite decimals extend to infinite decimals. This is true, but the proof is essentially the same as the proof of So, all these arguments are essentially circular reasoning. Nevertheless, the matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. discusses the argument that, in elementary school, one is taught that , so, ignoring all essential subtleties, "multiplying" this identity by gives . He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation ." Most undergraduate mathematics majors encountered by Byers feel that while is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption. Byers also presents the following argument. Students who did not accept the first argument sometimes accept the second argument, but, in Byers' opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. ,