The method is proved by following lemma:

Lemma 1. Let

be real numbers. For each

it is possible to approximate

simultaneously by rational numbers

*, *in the sense that

(

). In addition, if all the

are positive, then the

can be chosen to be positive, where

is replaced with

.

Proof. Let

* *be a positive integer satisfying

. Then by the pigeonhole principle, among the

points

(

), where

denotes the fractional part of

*,* there are at least two numbers

* * such that

for all

. Then

for some integers

. The statement is proved if we put

.

This lemma was used in the elementary proof of Hilbert's third problem.