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
Weyl's Equidistribution theorem for polynomials is equivalent to the claim that as . Though it is not the most easy way to prove this, let us deduce this theorem from the following Weyl's inequality.
Let be a rational number in lowest terms with . Weyl's Inequality is the bound:
If and are both large enough, and of the same order of magnitude, then the right-hand side gets small. The point is that the conditions on prevent one to apply this to arbitrary . However, Dirichlet's theorem tells us that arbitrary high satisfy the needed condition, so that Weyl's inequality implies .
Now the trick comes into play: the right-hand side in Weyl's inequality does not depend on , but only on . It therefore gives a uniform bound simultaneously for the sum and the sums computed using instead of .
For all , there is therefore a such that for all , . Then given any , one has
Then, for all , letting we get that . It follows that .
The principle of the trick is the following: dividing the domain (either geometric in optimization setting, of the domain of sumation or integration) can allow you to take an estimate on your sequence that holds at a point, and propagate it to all its multiples. Then, for all other terms you usually get a bounded additional error term, which is hopefully negligible. This can especially be applied when a compactness argument provides you with an inferior or superior limit.