Real Analysis/Section 1 Exercises

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Real Analysis
Exercises

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