She's proving t his theorem.
In the following, a sequence is an infinite sequence of 0's and 1's. Such a sequence is a function f : N -> {0,1} where N = {0,1,2,3, ...}.
Thus 10101010... is the function f with f(0) = 1, f(1) = 0, f(2) = 1, ... .
A sequence f is the characteristic function of the set {i : f(i) = 1}.
Thus 101010101... is the characteristic function of the set {0,2,4,6, ...}.
If X has characteristic function f(i), its complement has characteristic function 1 - f(i)