Quick description
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.
Prerequisites
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
.
General discussion
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.