Schnirelmann's theorem can be proved using Mann's theorem,
although Schnirelmann used the weaker inequality

where ,
,
and
is the Schnirelmann density. Let be the set of primes, together with 0 and
1, and let .
Using a sophisticated version of the inclusion-exclusion
principle, Schnirelmann showed that although , . By repeated applications of Mann's
theorem, the sum of copies of satisfies . Thus, if , the sum of copies of has Schnirelmann density
1, and so contains all positive integers.