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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