Real analysis/Section 1 Exercises

Exercises

1. Show that $-1<0$
2. Show that $\mathrm{\forall }x,y\in \mathbb{Z},-(xy)=(-x)y=x(-y)$
3. Show that $\mathrm{\forall }z\in \mathbb{Z},z^2\ge 0$
4. Show that $\mathrm{\forall }z,\in \mathbb{Z},z\ge 0\iff -z\le 0$
5. Show that $\mathrm{\forall }x,y,z\in \mathbb{Z},(x<0)\cap (y<z)\iff (xy>xz)$
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 $ℂ$ cannot be made into an ordered field.
9. Complete the proof of the square roots theorem by giving details for the case $x<1$.