Suppose $a$ is a real number.

1. If $a<0$ then $a$ is a negative number.
2. If $a>0$ then $a$ is a positive number.
3. If $a\le 0$ then $a$ is a non-positive number.
4. If $a> 0$ then $a$ is a non-negative number.

Properties

1. If $a>b$ , then $-a<-b$ . If $a<b$ , then $-a>-b$ .
2. If $a\ge b$ , then $-a\le -b$ . If $a\le b$ , then $-a\ge -b$ .

Proof. If $0<a$ , then adding $-a$ on both sides of the inequality gives $-a=-a+0<-a+a=0$ . This process can also be reversed.

Lemma 2   For any , either $a=0$ or $0<a^2$ .

Proof. Suppose $a\ne 0$ , then by trichotomy, we have either $0<a$ or $a<0$ , but not both. If $0<a$ , then $0=0\cdot a<a\cdot a=a^2$ . On the other hand, if $-(-a)=a<0$ , then $0<-a$ by the previous lemma. Then repeating the previous argument, $0 = 0\cdot(-a) < (-a)(-a)=a^2$ .

Three direct consequences follow:

Corollary 2   For any , $0<1+a^2$ .

Inequality for a converging sequence

1. If $a_i < b$ or $a_i \le b$ for some real number $b$ for each $i$ , then $a\le b$ .
2. If $a_i > b$ or $a_i \ge b$ for some real number $b$ for each $i$ , then $a\ge b$ .