# 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 $\alpha\le \beta$.
• msc/real_analysis_notes_by_syed_gul_shah/real_number_system