Ordering Properties

Samia CHALLAL

3 Ordering Properties

Order in $\mathbb{R}$ . For any two real numbers $a$ and $b$ , we have either

$a<b \qquad \hbox{ or }\qquad a>b \qquad \hbox{ or }\qquad a=b.$

We have

$\begin{eqnarray*} && \quad a < b \qquad\Longleftrightarrow\qquad a + c < b + c \\ \\ && \quad a < b \qquad\Longleftrightarrow\qquad \left\{ \begin{array}{ll} a c < b c & \hbox{ if } \quad c>0 \\ \\ a c > b c & \hbox{ if } \quad c<0 \end{array} \right. \\ \\ && \quad a < b\qquad \hbox{ and } \qquad b< c \qquad \Longrightarrow\qquad a< c \end{eqnarray*}$