inequalities for real numbers

Suppose $a$ is a real number.

1. If 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 $a\ge 0$ then $a$ is a non-negative number.

The first two inequalities are also called strict inequalities.
The second two inequalities are also called loose inequalities.

Properties

Suppose $a$ and $b$ are real numbers.

1. If $a>b$, then . If , then $-a>-b$.

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

Lemma 1.

  iff  .

Proof.

If , then adding $-a$ on both sides of the inequality gives .  This process can also be reversed.∎

Lemma 2.

For any $a\mathrm{\in }\mathrm{R}$, either $a\mathrm{=}\mathrm{0}$ or .

Proof.

Suppose  $a\ne 0$, then by trichotomy, we have either    or  , but not both.  If  ,  then  .  On the other hand, if  , then    by the previous lemma.  Then repeating the previous ,  .∎

Three direct consequences follow:

Corollary 1.

Corollary 2.

For any $a\mathrm{\in }\mathrm{R}$, .

Corollary 3.

There is no real solution for $x$ in the equation $\mathrm{1}\mathrm{+}{x}^{\mathrm{2}}\mathrm{=}\mathrm{0}$.

Inequality for a converging sequence

Suppose $a_0,a_1,\mathrm{\dots }$ is a sequence of real numbers converging to a realnumber $a$.

1. If 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 $a\ge b$.