In some situations the existence of a limit can be derived by using inferior or superior limit and suitably dividing a domain to use self-similarity.
Basic real analysis.
Example 1: Maximal density of a packing
Fix a compact domain in Euclidean space (for example, a ball). A packing is then a union of domains congruents to , with disjoint interiors. The density of a packing is defined as
where is the volume and is the square , when the limit exists.
The present trick shows that if is a packing contained in of maximal volume, then exists.
First, since , the superior limit exists. We want to prove that the inferior limit equals the superior one. Given any , there is an such that . Now for any integer the square can be divided into translates of . Each of these contains a packing of density greater than , so that for all . This rewrites as . By maximality, is non-decreasing in ; for all one can introduce and gets . It follows that
so that as soon as is large enough, . As a consequence, and we are done.
Example 2: Weyl's inequality and polynomial equidistribution
This example is taken from a mathoverflow question and answer.
Let be a polynomial with real coefficients, and irrational. Let