KöMaL 2020 B5120

The positive integers are coloured in the following manner: the colour of $a+b$ is always uniquely determined by the colours of $a$ and $b$; that is, if the colour of $a$ and $a'$ is the same, and the colour of $b$ and $b'$ is the same, then $a+b$ and $a'+b'$ also have the same colour. Prove that if there is a colour that is used more than once then the colouring becomes periodic from some number onwards.