Thread: analysis midterm

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.

Thanks.

Originally Posted by eigenvector11

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.

If so there exists rational so that thus that means is irrational. Q.E.D.

2. Prove that if a > 0, there is an integer n such that 1/n < a < n.

Given by Archimedean ordering there is nartual so that . Since by Archimedean ordering there is natural so that thus . Let then . Q.E.D.

3. For real numbers a and b, prove that if a <= b + 1/n for all positive integers n, then a <= b.

Assume not, then that means (by definition). Thus by Archimedian ordering with there is natural so that thus which is a contradiction. Q.E.D.

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.

The set (non-empty) has an upper bound, i.e. . Argue that this is the least upper bound.