In [[mathematics]], '''0.999...''' (also written as '''0.{{overline|9}}''', '''0.{{overset|.|9}}''', or '''0.(9)''') denotes the smallest number greater than every [[number]] in the sequence {{nowrap|(0.9, 0.99, 0.999, ...)}}. It can be proved that this number is{{spaces}}[[1]]; that is,

: <math>0.999... = 1.</math>

Let <math>x</math> be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, {{nowrap|0.(9)_{{{math|''n''}}} < <math> x </math> ≤ 1}}, for every {{tmath|1= n }}. By subtracting these inequalities from 1, one gets {{nowrap|0 ≤ 1 − <math> x </math> < {{tmath|1= \textstyle \frac{1}{10^n} }}}}.

Part of what this argument shows is that there is a [[least upper bound]] of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the [[axiom]]s of the [[real number system]] is the [[completeness axiom]], which states that every bounded sequence has a least upper bound.{{sfnp|Stillwell|1994|p=42}}{{sfnp|Earl|Nicholson|2021|loc="bound"}} This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations.{{sfnp|Rosenlicht|1985|p=27}} The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers has a least upper bound and that this least upper bound is equal to one.{{Citation neededsfnp|dateBauldry|2009|p=October 202447}}

}} One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying {{nowrap|1=0.999... = 1}} again. [[Tom Apostol]] concludes, "the fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."{{sfnp|Apostol|1974|p=12}}

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Commonly in [[secondary schools]]' mathematics education, the real numbers are constructed by defining a number using an integer followed by a [[radix point]] and an infinite sequence written out as a string to represent the [[fractional part]] of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the [[Real number#Axiomatic approach|real axioms]] after defining an [[equivalence relation]] over the set that defines {{nowrap|1=1 =_eq 0.999...}} as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version.{{sfnp|Li|2011}} WithIn thisother construction of the realswords, all proofs of the statementequality "{{nowrap|1=1 = 0.999... = 1}}" canholding betrue viewedis asa implicitlynecessary assumingcondition equalityfor whenstrings anyof operationsdigits areto performedbehave on theas real numbers should.{{Citation neededsfnp|date=October 2024Gowers|2001}}{{sfnp|Li|2011}}

With the rise of the [[Internet]], debates about 0.999... have become commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup {{mono|sci.math}} in the 1990s, arguing over 0.999... is described asbecame a "popular sport", and it iswas one of the questions answered in its [[FAQ]].{{sfnp|Richman|1999|p=396}}{{sfnp|de Vreught|1994}} The FAQ briefly covers {{tmath|1= \textstyle \frac{1}{3} }}, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A ''[[Slate (magazine)|Slate]]'' article reports that the concept of 0.999... is "hotly disputed on websites ranging from ''[[World of Warcraft]]'' message boards to [[Ayn Rand]] forums".<ref>{{cite websfnp|url=http://www.slate.com/blogs/how_not_to_be_wrong/2014/06/06/does_0_999_1_and_are_divergent_series_the_invention_of_the_devil.html|work=[[Slate (magazine)|Slate]]|author=Ellenberg, Jordan|title=Does {{nowrap|1=0.999... = 1}}? And Are Divergent Series the Invention of the Devil?|date=6 June 2014|author-link=Jordan Ellenberg|archive-date=8 August 2023|archive-url=https://web.archive.org/web/20230808022739/https://slate.com/human-interest/2014/06/does-0-999-1-and-are-divergent-series-the-invention-of-the-devil.html}}</ref> In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1: