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msc:real_analysis_notes_by_syed_gul_shahs:real_number_system

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- Theorem: There is no rational
*p*such that . - Theorem: Let
*A*be the set of all positive rationals*p*such that and let*B*consist of all positive rationals*p*such that 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*, there is an irrational number*u*such that . - Theorem: For every real number
*x*there is a set*E*of rational number such that . - Theorem: For every real and every integer , there is one and only one real
*y*such that . - The extended real numbers.
- Euclidean space.
- Theorem: Let . Then (i) (ii) .
- Question: Suppose then prove that (a) (b) .
- Question: If
*r*is rational and*x*is irrational then prove that and are irrational. - Question: If
*n*is a positive integer which is not perfect square then prove that is irrational number. - Question: Prove that is irrational.
- Question: Let
*E*be a non-empty subset of an ordered set, suppose is a lower bound of*E*and is an upper bound then prove that .

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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

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