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periodic continuous implies sequence is dense

Source: Miklos Schweitzer 2020, Problem 2

December 1, 2020
analysisfunctioncollege contestsMiklos Schweitzerreal analysis

Problem Statement

Prove that if f ⁣:R→Rf\colon \mathbb{R} \to \mathbb{R} is a continuous periodic function and α∈R\alpha \in \mathbb{R} is irrational, then the sequence {nα+f(nα)}n=1∞\{n\alpha+f(n\alpha)\}_{n=1}^{\infty} modulo 1 is dense in [0,1][0,1].
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