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 nonpositive number.

4.
If $a\ge 0$ then $a$ is a nonnegative 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 real number $a$.

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 $a\ge b$.
Title  inequalities for real numbers 

Canonical name  InequalitiesForRealNumbers 
Date of creation  20130322 13:58:16 
Last modified on  20130322 13:58:16 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  12 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 54C30 
Classification  msc 2600 
Classification  msc 12D99 
Related topic  SummedNumeratorAndSummedDenominator 
Defines  strict inequality 
Defines  inequality 