1.) Given that I is the set of irration numbers,
a.) Show that if a, b are in the rational numbers, then ab and a + b are elemenets of the rationals as well.
b.) Show that if a are in the rationals, and t is in the irrationals, then a + t is in the irrationals and at is in the irrationals as long as a does not equal 0.
c.) Part a says that the rationals is closed under addition and multilication. Are the irrationals closed under addition/multiplication? Given 2 irrationals (s and t), what are we able to say about s + t and st?
2.) Using the above, and using that the rationals are dense in the reals a - sqrt(2) and b - sqrt(2), prove that given any 2 real numbers where a < b, there exists an irrational number t that satisfies a < t < b.