We introduce an order "$ < $" on the field $\mathbb R$ and impose the the following it order properties for all $a,b,c\in\mathbb R$:

• Either $a < b$, or $a=b$, or $b < a$. (Law of Trichotomy)


• If $a < b$ and $b < c$, then $a < c$. (Transitivity)


• If $a < b$, then $a+c < b+c \;\forall\; c\in \mathbb R$. (Ordering after shifting)


• If $a < b$, then $ca < cb \;\forall\; c > 0$ and $ca > cb \;\forall\; c < 0$. (Ordering after scaling)

The well-ordering property cannot be proved just by the field and order properties of $\mathbb R$. Thus it is considered as an axiom or a principle (unless we assume mathematical induction).

We introduce a metric structure on the ordered field $\mathbb R$ by a distance function (metric) given by the absolute value function $|\cdot|$: $|x|=x$ if $x\geq 0$ and $|x|=-x$ if $x<0$.

The distance between two real numbers $x$ and $y$ is defined by $|x-y|$. Among known properties of $|\cdot|$, the following one is frequently used in real analysis:

Last edited by Dr. Mallik on