inequalities for real numbers

Suppose $a$ is a real number.

1. 1.

If $a<0$ then $a$ is a negative number.

2. 2.

If $a>0$ then $a$ is a positive number.

3. 3.

If $a\leq 0$ then $a$ is a non-positive number.

4. 4.

If $a\geq 0$ 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. 1.

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

2. 2.

If $a\geq b$, then $-a\leq-b$. If $a\leq b$, then $-a\geq-b$.

Lemma 1.

$0  iff  $-a<0$.

Proof.

If $0, 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 $a\in\mathbbmss{R}$, either $a=0$ or $0.

Proof.

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

Three direct consequences follow:

Corollary 1.

$0<1$

Corollary 2.

For any $a\in\mathbbmss{R}$, $0<1+a^{2}$.

Corollary 3.

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

Inequality for a converging sequence

Suppose $a_{0},a_{1},\ldots$ is a sequence of real numbers converging to a real number $a$.

1. 1.

If $a_{i} or $a_{i}\leq b$ for some real number $b$ for each $i$, then $a\leq b$.

2. 2.

If $a_{i}>b$ or $a_{i}\geq b$ for some real number $b$ for each $i$, then $a\geq b$.

Title inequalities for real numbers InequalitiesForRealNumbers 2013-03-22 13:58:16 2013-03-22 13:58:16 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 54C30 msc 26-00 msc 12D99 SummedNumeratorAndSummedDenominator strict inequality inequality