Thanks to this choice, we can find a sequence such that and , hence we have shown that we don't have the part after the "if and only if".
I think you mean q instead of F(p). So essentialy that is chosen since 1/n-> 0 (allowing x_n to be arbitarily close to 0) and x_n consists of all 'rogue' values of x is so far as they make F(x) more than epsilon away from q. Is this correct?