Rainbow Colourings of Chains

We prove that if \(m\) be a positive integer and \(X\) is a totally ordered set, then there exists a function \(\phi: X \to \{1,\ldots,m\}\) such that, for every interval \(I\) in \(X\) and every positive integer \(r \leq |I|\), there exist elements \(x_1 < x_2 < \cdots < x_r\) of \(I\) such that \(\phi(x_{i+1}) \equiv \phi(x_{i}) + 1 \pmod{m}\) for \(i=1,\ldots,r-1\).