Division algorithm and Kronecker’s approximation theorem

Abstract

Using division algorithm and basic notions of convergence of sequences in real–line, we prove that a real number $$\theta$$ is irrational if and only if there is an eventually nonconstant sequence $$\{p_n\theta +q_n\}$$ converging to 0, where $$p_n$$ and $$q_n$$ are integers for each natural number n. This approach leads to alternative proofs of weaker versions of the classical Dirichlet and Kronecker approximation theorems in number theory.