# Chapter 01 - Real Number System

### Contents & Summary

- 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 $\alpha\le \beta$.

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