Real numbers, sequences and series: part VI

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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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