Chapter 01 - Real Number System

Theorem: There is no rational p such that $p^2=2$.

Theorem: Let A be the set of all positive rationals p such that $p^2>2$ and let B consist of all positive rationals p such that $p^2<2$ then A contain no largest member and B contains no smallest member.

Order on a set.

Ordered set.

Bound.

Least upper bound (supremum) & greatest lower bound (infimum).

Least upper bound property.

Theorem: An ordered set which has the least upper bound property has also the greatest lower bound property.

Field.

Proofs of axioms of real numbers.

Ordered field.

Theorems on ordered field.

Existence of real field.

Theorem: (a) Archimedean property (b) Between any two real numbers there exits a rational number.

Theorem: Given two real numbers x and y, $x<y$ there is an irrational number u such that $x<u<y$.

Theorem: For every real number x there is a set E of rational number such that $x=\sup E$.

Theorem: For every real $x>0$ and every integer $n>0$, there is one and only one real y such that $y^n=x$.

The extended real numbers.

Euclidean space.

Theorem: Let $\underline x,\underline y\in \mathbb{R}^n$. Then (i) $\|\underline x^2\|=\underline x\cdot \underline x$ (ii) $\|\underline x\cdot \underline y\|=\|\underline x\| \|\underline y\|$.

Question: Suppose $\underline x,\underline y, \underline z\in \mathbb{R}^n$ then prove that (a) $\left\| {\,\underline x + \underline y \,} \right\| \le \left\| {\,\underline x \,} \right\| + \left\| {\,\underline y \,} \right\|$ (b) $\left\| {\,\underline x - \underline z \,} \right\| \le \left\| {\,\underline x - \underline y \,} \right\| + \left\| {\,\underline y - \underline z \,} \right\|$.

Question: If r is rational and x is irrational then prove that $r+x$ and are $rx$ irrational.

Question: If n is a positive integer which is not perfect square then prove that $\sqrt{n}$ is irrational number.

Question: Prove that $\sqrt{12}$ is irrational.

Question: Let E be a non-empty subset of an ordered set, suppose $\alpha$ is a lower bound of E and $\beta$ is an upper bound then prove that $\alpha\le \beta$.