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inequalities for real numbers (Definition)

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.
The first two inequalities are also called strict inequalities.

Properties

Suppose $ a$ and $ b$ are real numbers.
  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$.
Lemma 1   $ 0<a$ iff $ -a<0$.
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. $ \qedsymbol$
Lemma 2   For any $ a\in \mathbbmss{R}$, 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$. $ \qedsymbol$

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. 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$.



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"inequalities for real numbers" is owned by mathcam. [ full author list (4) | owner history (1) ]
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See Also: summed numerator and summed denominator

Also defines:  strict inequality, inequality

Attachments:
positive (Definition) by pahio
quadratic inequality (Topic) by pahio
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Cross-references: sequence, equation, solution, consequences, trichotomy, sides, iff, real number
There are 230 references to this entry.

This is version 7 of inequalities for real numbers, born on 2003-09-26, modified 2006-03-04.
Object id is 4742, canonical name is InequalityForRealNumbers.
Accessed 20148 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 26-00 (Real functions :: General reference works )
 54C30 (General topology :: Maps and general types of spaces defined by maps :: Real-valued functions)
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