Theorems in real analysis | Sets of real numbers
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: 1. * Every interval in the sequence is contained in the previous one ( is always a subset of ). 2. * The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ). In other words, the left bound of the interval can only increase, and the right bound can only decrease. Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and surcumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi. The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval (thus, for all ). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers). (Wikipedia).
Closed Intervals, Open Intervals, Half Open, Half Closed
00:00 Intro to intervals 00:09 What is a closed interval? 02:03 What is an open interval? 02:49 Half closed / Half open interval 05:58 Writing in interval notation
From playlist Calculus
Interval of Convergence (silent)
Finding the interval of convergence for power series
From playlist 242 spring 2012 exam 3
A beat is an interference pattern between two sounds of slightly different in frequencies You can download this app or a similar app on two devices and TRY it at home Enjoy!!!
From playlist Beats
Amazing science experiment-Demonstrating beat frequency
A beat is an interference pattern between two sounds of slightly different in frequencies You can download this app or a similar app on two devices and TRY it at home Enjoy!!!
From playlist Beats
Amazing science experiment-Demonstrating beat frequency
A beat is an interference pattern between two sounds of slightly different in frequencies You can download this app or a similar app on two devices and TRY it at home Enjoy!!!
From playlist Beats
This video defines set-builder notation and compares it to interval expressed graphically, using interval notation, and using inequalities. Site: http://mathispower4u.com
From playlist Using Interval Notation
This is an old video. See StatsMrR.com for access to hundreds of 1-3 minute, well-produced videos for learning Statistics. In this older video: Understanding and constructing a confidence interval for one mean when the population standard deviation is known
From playlist Older Statistics Videos and Other Math Videos
Nested Interval Property and Proof | Real Analysis
We introduce and prove the nested interval property, or nested interval theorem, or NIP, whatever you like to call it. This theorem says that, given a sequence of nested and closed intervals, that is, closed intervals J1, J2, J3, and so on such that each Jn contains Jn+1, this infinite seq
From playlist Real Analysis
Proving Bolzano-Weierstrass with Nested Interval Property | Real Analysis
We prove the Bolzano Weierstrass theorem using the Nested Interval Property. The Bolzano-Weierstrass theorem states every bounded sequence has a convergent subsequence. We will construct a subsequence by bounding our sequence between M and -M, then creating an infinite sequence of nested i
From playlist Real Analysis
What is a Manifold? Lesson 4: Countability and Continuity
In this lesson we review the idea of first and second countability. Also, we study the topological definition of a continuous function and then define a homeomorphism.
From playlist What is a Manifold?
Real Analysis Ep 18: Compact sets
Episode 18 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about compact sets. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/courses/2020f3371/ Chris Staecker webpage: http://faculty.fairfield
From playlist Math 3371 (Real analysis) Fall 2020
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Episode 4 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about the Archimedean property of the real numbers. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/courses/2020f3371/ Chris Staecker w
From playlist Math 3371 (Real analysis) Fall 2020
Real Analysis Ep 6: Countable vs uncountable
Episode 6 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about countable and uncountable sets, Cantor's theorem, and the continuum hypothesis. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/c
From playlist Math 3371 (Real analysis) Fall 2020
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Episode 3 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about the completeness axiom for the real numbers. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/courses/2020f3371/ Chris Staecker we
From playlist Math 3371 (Real analysis) Fall 2020
Real Analysis | The uncountability of ℝ
We prove that the real numbers are uncountable by way of the nested interval property. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/mathematics/ Research Ga
From playlist Real Analysis
Real Analysis | Nested compact sets.
We prove a generalization of the nested interval theorem. In particular, we prove that a nested sequence of compact sets has a non-empty intersection. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Person
From playlist Real Analysis
Lesson: Calculate a Confidence Interval for a Population Proportion
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From playlist Confidence Intervals
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From playlist Medical Statistics
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From playlist Real Analysis