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msc:real_analysis_notes_by_syed_gul_shahs:real_number_system

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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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msc/real_analysis_notes_by_syed_gul_shahs/real_number_system.txt · Last modified: 2014/08/26 16:35 by Administrator