# Real numbers, sequences and series: part VI

No study is complete without solving problems on your own. Below are the exercises related to part V.

Exercises.

Using the properties of real numbers, show that

1) $\alpha <0$ or $\alpha=0$ or $\alpha>0$.

2) If $\alpha \neq 0$, $(-\alpha)^{2}=(\alpha)^{2}>0$.

3) if $0, then $\frac{1}{b} \leq \frac{1}{a}$.

4) If $0 \leq a,b$, then $(1-a)(1-b) \geq 1-a-b$.

5) If $0, then $a^{n}<1$ for any positive integer n.

6) For $0, $a^{n} implies $a.

7) If $0 \leq a$, then $(1+a)^{n}\geq 1+na$, and $(1-a)^{n} \geq 1-na$, and

8) If $0, then $(1+a)^{n}<\frac{1}{(1-a)^{n}}<\frac{1}{1-na}$

9) Every set $A \subseteq \Re$ that is bounded below admits a greatest lower bound.

More later,

Nalin Pithwa

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