Supremum Study guides, Revision notes & Summaries
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Analysis Notes - Supremum and Infimum
- Lecture notes • 9 pages • 2023
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The notes cover definitions, propositions and examples relating to the supremum and infimum of a set and how to calculate these quantities.
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Real Analysis I Maths Notes
- Lecture notes • 27 pages • 2024
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Lecture notes for the module 'MAT1032 Real Analysis I' in the Mathematics course. 
Topics covered in the notes include: 
- Bounds, Supremum, Infimum 
- Axiom of Completeness 
- Cauchy Sequences and Series 
- Inequalities 
- Limits and Covergence 
- Logic and Quantifiers 
- Sets and Notations 
- Types of Functions 
- Types of Numbers 
 
Definitions in pink, formulae and equations in green, examples in blue.
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Stanford UniversitySTATS 231hw0-solutions
- Exam (elaborations) • 10 pages • 2021
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Homework 0 solutions 
CS229T/STATS231 
1. Linear algebra (0 points) 
a (dual norm of L1 norm) The L1 norm k · k1 of a vector v 2 Rn is defined as 
kvk1 = 
nX i 
=1 
jvij: (1) 
The dual norm k · k∗ of a norm k · k is defined as 
kvk∗ = sup 
kwk≤1 
(v · w): (2) 
Compute the dual norm of the L1 norm. (Here v · w denotes the inner product between v and w: v · w , 
Pn i=1 viwi) 
Solution: 
We will prove that 
sup 
kwk1≤1 
(v · w) = max 
i2[n] 
vi = kvk1 (3) 
which implies that the dua...
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Axiom of Completeness and Consequences
- Lecture notes • 3 pages • 2023
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Introduce Axiom of Completeness, the bed rock of advanced calculus, where every nonempty bounded above subset of R has a lower upper bound (supremum), and every nonempty bounded below subset of R has a greatest lower bound (infimum). Introduce the consequences of the Axiom of Completeness, with the Nested Interval Property and the Archimedean Property.
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Exam (elaborations) TEST BANK FOR Understanding Analysis 2nd Edition By Stephen Abbott (Instructors' Solution Manual)
- Exam (elaborations) • 156 pages • 2021
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Exam (elaborations) TEST BANK FOR Understanding Analysis 2nd Edition By Stephen Abbott (Instructors' Solution Manual) 
Contents 
Author’s note v 
1 The Real Numbers 1 
1.1 Discussion: The Irrationality of 
p 
2 . . . . . . . . . . . . . . . . . 1 
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 6 
1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 8 
1.5 Cantor’s Theore...
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Comprehensive Guide to Mathematical Analysis: Foundations, Limits, and Continuity
- Lecture notes • 4 pages • 2023
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Abstract: 
This document serves as a comprehensive guide to the fundamental concepts of mathematical analysis. It provides a thorough exploration of the foundational principles, limits, and continuity, equipping readers with a strong understanding of these crucial mathematical concepts. 
 
Key Topics Covered: 
 
Introduction to Mathematical Analysis 
Sets and their Properties 
Axiom of Completeness and Real Numbers 
Sequences and their Convergence 
Limits: Definition and Types 
One-sided Limits ...
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We can easily get the idea about it by learning the content in it.
- Lecture notes • 4 pages • 2022
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In real analysis, the monotone convergence theorem states that if a sequence increases and is bounded above by a supremum, it will converge to the supremum; similarly, if a sequence decreases and is bounded below by an infimum, it will converge to the infimum.
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We can get the idea about lub by studying the theorem and definition of it.
- Lecture notes • 4 pages • 2022
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In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property)[1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set {displaystyle mathbb {Q} }mathbb {Q} of all rational numbers wit...
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Limit of Sequence
- Lecture notes • 2 pages • 2023
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Give an epsilon proof for the limit of a sequence, where in more detail the partial sum of a series and its convergence is explained and tested with the p-series test. Then, introduce the Cauchy sequence and the monotone convergence theorem.
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Consequences of Axiom of Completeness
- Lecture notes • 1 pages • 2023
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Explain the proof for the nested interval property, and also the proofs for the Archimedean Property, which is split into two parts.
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