The is like an epsilon. If a set is not open, then there is a point in that set such that every -neighborhood around the point contains points not in the set, correct? In this case, we're considering , which by hypothesis is not open. The point But now the author wants to create a sequence of points in a sequence of index-able neighborhoods, so instead of using an he uses the equivalent sequence in order to be able to index them. The sequence and hence is not in by construction.
Does that make sense?