MathDB

Let

\alpha

and

\beta

be irrational numbers with the property that

\frac{1}{\alpha} +\frac{1}{\beta}=1

Let

\{a_n\}

and

\{b_n\}

be the sequences given by

a_n= \lfloor n\alpha \rfloor

and

b_n= \lfloor n\beta \rfloor

respectively. Prove that the sequences

\{ a_n\}

and

\{ b_n \}

has no term in common and cover all the natural numbers.
I know this theorem from long ago, but forgot the proof of it. Can anybody help me with this?

Problem Statement