Sorry for all the questions, but I have a midterm coming up and I'm having problems with these questions:
1. Prove that if a < b, then there is an irrational number x such that
a < x < b. Hint: first show that if q not equal to zero is rational, then q+ root(2) and q*root(2) are irrational.
2. Prove that if a > 0, there is an integer n such that 1/n < a < n.
3. For real numbers a and b, prove that if a <= b + 1/n for all positive integers n, then a <= b.
4. For each real number a, let the set S_a be the set of all rational numbers less than a. Prove that for each real number a, sup S_a = a.