proof of Beatty’s theorem
We define and . Since and are irrational, so are and .
It is also the case that for all and , for if then would be rational.
Choose integer. Let be the number of elements of less than .
So there are elements of less than and likewise elements of .
and summing these inequalities gives which gives that since is integer.
The number of elements of lying in is then .
|Title||proof of Beatty’s theorem|
|Date of creation||2013-03-22 13:18:58|
|Last modified on||2013-03-22 13:18:58|
|Last modified by||lieven (1075)|