Real numbers, sequences and series: part VI

Posted by Nalin Pithwa

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 $0<a\leq b$ , 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 $0<a<1$ , then $a^{n}<1$ for any positive integer n.

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

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

8) If $0<a<\frac{1}{n}$ , 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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