# Real Analysis/Section 1 Exercises/Hints

### Exercises

1. Show that $-1<0\$
2. Show that $\forall x,y\in\mathbb{Z},\ -(xy)=(-x)y=x(-y)$
3. Show that $\forall z\in\mathbb{Z},\ z^2\ge0$
4. Show that $\forall z\in\mathbb{Z},\ z\ge0\Leftrightarrow-z\le0$
5. Show that $\forall x,y,z\in\mathbb{Z},\ (x<0)\cap(yxz)$
6. Let $p$ be any prime. Show that $\sqrt{p}$ is irrational.
7. Complete the proofs of the simple results given above.
8. Show that the complex numbers $\mathbb C$ cannot be made into an ordered field.
9. Complete the proof of the square roots theorem by giving details for the case $x<1$.
10. Suppose A is a non-empty set of real numbers that is bounded above and let s = sup A. Show that if s is not in A, then for any ε > 0, there exists an element a in A such that s − ε < a < s.

8. The only fact you need use about $\mathbb C$ is that it contains a square root of $-1$.
9. In the general case you will probably want to divide by your prospective square root, as in the part of the proof which was given, so you might want to treat the case $x=0$ separately.