Suppose that is -coloured.
Then there exists an infinite monochromatic set.
Proof
We prove this by induction on .
The case is trivial.
Assume the statement for and let be a colouring.
Construct the sequences for all .
Let and .
Consider the colouring
given by
Let be an infinite monochromatic subset of under
(given by the induction hypothesis)
Thus we have a sequence of distinct
and some colours such that
whenever .
But infinitely many of will be the same,
so choose this subsequence for the infinite monochromatic set.