At the beginning of the prove, there is this sentence:
There are $k = (2m)^m$ possible sequences of $m$ examples from $C$.
From other source:
If our input space consists of $2m$ points and our training data consists of a sequence of $m$ points, then there are $k = (2m)^m$ possible sequences.
As I understood, in every of the $2m$ spots I can place some of my $m$ examples (and they can be repeated).
How do this sequences look like? Let C = [a,b] (2 points), then I want to take the simplest example, let $m=1$, input space is 2 and $k=2$, so we should have 2 sequences but its impossible to have 2 sequences with $m=1$. Or are the 2 sequences ab and aa or bb and ab, or something else? EDIT: In this case 2 sequences should be just a and b I think.
It is the other way around: in every one of $m$ spots you can place one of $2m$ possibilities.
It is hard to provide further help without some more context.