I'd be happy if someone could justify the union bound step in this proof in a more precise manner.
At the top of page 5 in this article http://www.gatsby.ucl.ac.uk/~risi/courses/ML2/2008/VC.pdf, it is stated:
The union bound is applied only on a finite set of hypotheses because two hypythesis h and h' only need to be counted as distinct if they differ on one of the samples in S or S'. But if S and S' change, so do the distinct hypothese that we bound on. The aforementioned h and h' may become or cease to be distinct. To conclude, the hypothese to apply the bound over, change with each sample.
How exactly can this step be justified? What are the events(subsets of the sample space) in the union bound?