Editing Almost all (section)

===Meaning in number theory===
{{further|Asymptotically almost surely}}
In [[number theory]], "almost all positive integers" can mean "the positive integers in a set whose [[natural density]] is 1". That is, if <var>A</var> is a set of positive integers, and if the proportion of positive integers in ''A'' below <var>n</var> (out of all positive integers below <var>n</var>) [[limit of a sequence|tends to]] 1 as <var>n</var> tends to infinity, then almost all positive integers are in <var>A</var>.{{r|Hardy1|Hardy2}}{{r|Weisstein|group=sec}} 

More generally, let <var>S</var> be an infinite set of positive integers, such as the set of even positive numbers or the set of [[prime number|primes]], if <var>A</var> is a subset of <var>S</var>, and if the proportion of elements of <var>S</var> below <var>n</var> that are in <var>A</var> (out of all elements of <var>S</var> below <var>n</var>) tends to 1 as <var>n</var> tends to infinity, then it can be said that almost all elements of <var>S</var> are in <var>A</var>.

Examples:
* The natural density of [[cofinite set]]s of positive integers is 1, so each of them contains almost all positive integers.
* Almost all positive integers are [[composite number|composite]].{{r|Weisstein|group=sec}}{{refn |group=proof |The [[prime number theorem]] shows that the number of primes less than or equal to <var>n</var> is asymptotically equal to <var>n</var>/ln(<var>n</var>). Therefore, the proportion of primes is roughly ln(<var>n</var>)/<var>n</var>, which tends to 0 as <var>n</var> tends to [[infinity]], so the proportion of composite numbers less than or equal to <var>n</var> tends to 1 as <var>n</var> tends to infinity.{{r|Hardy2}}}}
* Almost all even positive numbers can be expressed as the sum of two primes.{{r|Courant|page=489}}
* Almost all primes are [[twin prime#Isolated prime|isolated]]. Moreover, for every positive integer {{mvar|g}}, almost all primes have [[prime gap]]s of more than {{mvar|g}} both to their left and to their right; that is, there is no other prime between {{math|''p'' − ''g''}} and {{math|''p'' + ''g''}}.{{r|Prachar}}