81. Control Theory And Engineering Links Int Federation of Automatic Control (IFAC2000); Virtual Library of Control; International Linear algebra Society; Comments or suggestions? Email www@theorem.net. http://www.theorem.net/control.html

Control Theory and Engineering Links from Theorem.Net On-Line Books or Chapters (free access) Books Freeware: Applets, Toolboxes, etc ... Control Groups Around the World Freeware: Applets, Toolboxes, etc Demos from Johns Hopkins Control System Labs Online (U of Tennessee at Chattanooga) Control Tutorials from Michigan SciLab (a Matlab-like free package) Matlab or Matlab on-line tutorials Resources Search for papers and information: CiteSeer Los Alamos (a few free searches) NA-net: search interface or front page Nonlinear Control Abstracts IEEE Bibliographies on-line (must be IEEE member to use) Search AMS Database France mirror site) (must subscribe to use) Elsevier journals (click "Journal TOCs and abstracts") Information Services and Mailing Lists: Electronic Newsletter (Eletter) on Systems, Control, and Signal Processing Systems and Control Web Site (provided by UT Dallas) Control Virtual Library (Caltech mirror) NA-net UK Nonlinear News , and direcly to ( book reviews Mathematics Virtual Library Control (and related) Organizations: IFAC (International Federation of Automatic Control) ( USA mirror AACC (American Automatic Control Council) EUCA (European Union Control Association) UKACC (United Kingdom Automatic Control Council) ERCIM (European Research Consortium for Informatics and Mathematics) IEEE activity groups: Control Systems Society conferences Discrete Event Systems Hybrid Systems ... Dynamical Systems group Funding Agencies: NSF US Natl.Sci.Foundation

82. GeoSci 236: The Fundamental Theorem Of Linear Algebra GeoSci 236 The Fundamental theorem of Linear algebra. Gidon Eshel 491 Hinds Dept. Figure 1 The forward problem (the fundamental theorem of linear algebra). http://geosci.uchicago.edu/~gidon/geosci236/fundam/

GeoSci 236: The Fundamental Theorem of Linear Algebra Gidon Eshel 491 Hinds Dept. of the Geophysical Sciences, 5734 S. Ellis Ave., The Univ. of Chicago, Chicago, IL 60637 geshel@midway.uchicago.edu Figure 1: The forward problem (the fundamental theorem of linear algebra). A 's domain is the upper-left space, while its range is the lower-right one. In the domain, A 's row-space is shown in red , while its nullspace in blue . A generic vector comprising both a row-space and a nullspace components is the vector on which A operates, mapping it onto the adjoint space (lower-right). In the latter space, the shown b comprises components from A 's range (column-space) and left nullspace Figure 1 represents the operation of a matrix on a vector (the upper-left space). That is, it shows schematically what happens when an arbitrary vector from 's domain (the space corresponding dimensionally to 's row dimension N ) is mapped by onto the range space (the space corresponding dimensionally to 's column dimension M ). Hence the schematic shows what happens to from the upper-left space as transforms it to the range, the lower-right space. Put differently, this schematic represents the

83. Commalg.org - The Center For Commutative Algebra polynomial rings over a field, April 1, 2004 Hakopian The Multivariate Fundamental theorem of algebra and algebraic Geometry, March 31, 2004 Bothmer Generic http://www.commalg.org/preprints/

@import "/style.css"; Preprints Below are the most recent additions to our files. Click on a name to see the announcement. Click here for previous months' preprints. Holm-Jorgensen : Cohen-Macaulay injective, projective, and flat dimension; Semi-dualizing modules and related Gorenstein homological dimensions, May 28, 2004 Baker-Richter Dibaei-Yassemi : Associated primes and cofiniteness of local cohomology modules, May 27, 2004 Weimann : La trace via le calcul residuel: une nouvelle version du theoreme d'Abel-inverse, formes abeliennes, May 27, 2004 Vaccarino : The ring of multisymmetric functions, May 27, 2004 Brenner : A linear bound for Frobenius powers and an inclusion bound for tight closure, May 26, 2004 Brenner-Hein : Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz fnctions, May 26, 2004 Drensky : Invariants of unipotent transformations acting on noetherian relatively free algebras , May 25, 2004 Aslaksen-Drensky-Sadikova Eisenbud-Huneke-Ulrich : The Regularity of Tor and Graded Betti Numbers, May 20, 2004 : Bigraded structures and the depth of blow-up algebras, May 20, 2004

84. Schiller Institute -Pedagogy - Gauss's Fundamental Theorem Of A;gebra Carl Gauss s Fundamental theorem of algebra. His Declaration of Independence. by Bruce Director April, 2002. Carl Gauss s Fundamental theorem of algebra. http://www.schillerinstitute.org/educ/pedagogy/gauss_fund_bmd0402.html

Home Search About Fidelio ... Dialogue of Cultures SCHILLER INSTITUTE Carl Gauss's Fundamental Theorem of Algebra H is Declaration of Independence by Bruce Director April, 2002 To List of Pedagogical Articles To Diagrams Page To Part II Lyndon LaRouche on the Importance of This Pedagogy ... Related Articles Carl Gauss's Fundamental Theorem of Algebra Disquisitiones Arithmeticae Nevertheless, he took the opportunity to produce a virtual declaration of independence from the stifling world of deductive mathematics, in the form of a written thesis submitted to the faculty of the University of Helmstedt, on a new proof of the fundamental theorem of algebra. Within months, he was granted his doctorate without even having to appear for oral examination. Describing his intention to his former classmate, Wolfgang Bolyai, Gauss wrote, "The title [fundamental theorem] indicates quite definitely the purpose of the essay; only about a third of the whole, nevertheless, is used for this purpose; the remainder contains chiefly the history and a critique of works on the same subject by other mathematicians (viz. d'Alembert, Bougainville, Euler, de Foncenex, Lagrange, and the encyclopedists ... which latter, however, will probably not be much pleased), besides many and varied comments on the shallowness which is so dominant in our present-day mathematics." In essence, Gauss was defending, and extending, a principle that goes back to Plato, in which only physical action, not arbitrary assumptions, defines our notion of magnitude. Like Plato, Gauss recognized it were insufficient to simply state his discovery, unless it were combined with a polemical attack on the Aristotelean falsehoods that had become so popular among his contemporaries.

85. Abstract Algebra Notes (PostScript) These are notes for Abstract algebra II. They were revised during the Fall, 1999 term. action.ps (7 pages, 113,662 bytes) Group actions; Burnside s theorem. http://www.millersv.edu/~bikenaga/absalg/absanote.html

Abstract Algebra Notes (PostScript) These are links to PostScript files containing notes for various topics in abstract algebra. How do I view/print the notes? How may I use/distribute the notes? How did you write them? These are notes for Abstract Algebra I; they were revised during the Fall, 1999 term. ringint.ps (2 pages, 35,802 bytes): Axioms for the ring of integers; well-ordering. div.ps (10 pages, 274,812 bytes): Divisibility, greatest common divisor, the Euclidean algorithm. prime.ps (3 pages, 44,277 bytes): Prime numbers. modular.ps (6 pages, 150,456 bytes): Arithmetic in the integers mod n; congruences. groups.ps (14 pages, 275,279 bytes): The axioms for a group; basic properties and examples. matgrp.ps (3 pages, 40,539 bytes): Groups of matrices. unitzn.ps (3 pages, 44,922 bytes): The group of units in the integers mod n; Fermat's theorem. subgroup.ps (6 pages, 113,670 bytes): Subgroups. cyclic.ps (6 pages, 84,587 bytes): Cyclic groups. These are notes for Abstract Algebra I; they were revised during the Fall, 1998 term. perm.ps (8 pages, 139K):

86. Tulane Math Graduate Algebra Qualifying Exam Syllabus Euclidean rings, PID s. Rings with chain conditions, Hilbert basis theorem. Prime fields. Algebraic and transcendental extensions. Algebraically closed fields. http://www.math.tulane.edu/graduate/qualifying/algebra.html

Home Office hours Weekly activities Find people ... Undergraduate Graduate Courses Summer research Lectures... Algebra syllabus Algebra Analysis Scientific Computation Topology Go to... Graduate studies Description Requirements Ph.D requirements ... Qualifying exams Topics Elementary Number Theory ([1]) Divisibility, prime numbers and the number of prime numbers. Unique factorization. Congruences, congruence theorems of Fermat and Euler. Euler's phi-function. Linear Diophantine equations, Pythagorean triples. Groups ([3, 4]) Rings and Fields ([3, 4]) Rings, ideals. Prime and maximal ideals in commutative rings. Field of quotients. Polynomial rings, matrix rings, boolean rings. Euclidean rings, PID's. Rings with chain conditions, Hilbert basis theorem. Prime fields. Algebraic and transcendental extensions. Algebraically closed fields. Finite fields. Galois groups, solvability of algebraic equations, constructions with straight edge and compass. Modules ([2, 3, 4])

87. Theorem EQ-2: Robbins Algebra, Boolean next up previous Next theorem EQ3 On Ternary Up Summary of Otter Outputs Previous theorem EQ-1 The Commutator theorem EQ-2 Robbins algebra, 2 2 Boolean. http://www-fp.mcs.anl.gov/~lusk/papers/contest/node18.html

Next: Theorem EQ-3: On Ternary Up: Summary of Otter Outputs Previous: Theorem EQ-1: The Commutator Theorem EQ-2: Robbins Algebra, 2#2 Boolean Karen D. Toonen

88. Theorem EQ-5: On Wajsberg Algebra theorem EQ5 On Wajsberg algebra. - OTTER 2.2, July 1991 - The job began on altair.mcs.anl.gov, Thu Jun 4 173143 1992 The command was otter22 . http://www-fp.mcs.anl.gov/~lusk/papers/contest/node21.html

Next: Conclusion Up: Summary of Otter Outputs Previous: Theorem EQ-4: Group Theory Theorem EQ-5: On Wajsberg Algebra Karen D. Toonen

89. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Preliminaries; Operations on Polynomials; Substitution in Polynomials; Fundamental theorem of algebra. Bibliography. http://mizar.uwb.edu.pl/JFM/Vol12/polynom5.html

Journal of Formalized Mathematics Volume 12, 2000 University of Bialystok Association of Mizar Users Fundamental Theorem of Algebra Robert Milewski University of Bialystok This work has been partially supported by TYPES grant IST-1999-29001. MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Preliminaries Operations on Polynomials Substitution in Polynomials Fundamental Theorem of Algebra Bibliography 1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences Journal of Formalized Mathematics 2] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics 3] Grzegorz Bancerek. The ordinal numbers Journal of Formalized Mathematics 4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics 5] Czeslaw Bylinski. Binary operations Journal of Formalized Mathematics 6] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 7] Czeslaw Bylinski. Functions from a set to a set Journal of Formalized Mathematics 8] Czeslaw Bylinski.

90. "ALGEBRA" Remainder Theorem & Synthetic Division For Polynomials algebra remainder theorem synthetic division for polynomials. http://www.algebrahelp.com/messageboards/messages/4410.html

"ALGEBRA" remainder theorem & synthetic division for polynomials Follow Ups Post Followup Algebra.Help Message Board Posted by syjo1@myexcel.com (204.30.143.234) on August 27, 2003 at 04:39:45: can someone PLEASE assist me with these 2 problems? for each polynomial, I am to use the remainder theorem and synthetic dividion to find f(k) k= -2; f(x)=x^2+5x+6 k=2; f(x)=2x^3-3x^2-5x+4 thank you so very much! The person who posted this message also included links to the following area(s) on algebrahelp.com: Polynomials (Order of Operations) Calculator Resources Formulas Follow Ups: This site explains how to do this stuff. Post A Followup Message: Follow Ups Post Followup Algebra.Help Message Board

91. Robbins Algebras Are Boolean A web text by William McCune describing the solution of this problem by a theoremproving program, with input files and the proofs. http://www-unix.mcs.anl.gov/~mccune/papers/robbins/

Robbins Algebras Are Boolean William McCune Automated Deduction Group Mathematics and Computer Science Division Argonne National Laboratory Posted on the Web October 15, 1996. Last updated September 24, 2003. These Web pages contain some information on the solution of the Robbins problem. A paper on this topic appears in the Journal of Automated Reasoning [W. McCune, "Solution of the Robbins Problem", JAR 19(3), 263276 (1997)]. Here is a preprint . The JAR paper has simpler proofs than the ones below on this page. Here are the input files and proofs corresponding to the JAR paper A draft of a press release , intended for a wider audience, is also available. Introduction The Robbins problem-are all Robbins algebras Boolean?-has been solved: Every Robbins algebra is Boolean. This theorem was proved automatically by EQP , a theorem proving program developed at Argonne National Laboratory. Historical Background In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra: x + y = y + x. [commutativity] (x + y) + z = x + (y + z). [associativity] n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation] Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one [5]:

92. Renate Schmidt: Home Page University of Manchester Modal logic, resolution theorem proving, resolution decision problems, relation algebras, Peirce algebras and knowledge representation. http://www.cs.man.ac.uk/~schmidt/

Dr.-Ing. Renate A. Schmidt Department of Computer Science University of Manchester Oxford Rd, Manchester M13 9PL, UK. Tel: +44 (0)161 275 6163, Fax: +44 (0)161 275 6204. Mobile: +44 (0)776 193 5696. Email: schmidt@cs.man.ac.uk Kilburn Building (formerly Computer Science Building), Room: 2.42. Research group: Formal Methods (FM) Upcoming events Advances in Modal Logics 2004 , 9-11 September 2004, Manchester Workshop of Tarski EU COST Action 274 , 12-13 September 2004, Manchester. Publications Publications ordered by publication date Publications ordered by theme Research interests Research projects Professional activities I am a member of the editorial board of the Journal of Applied and Non-Classical Logic . If you would like me to handle your paper please either send me your paper by email (PostScript or PDF format preferred) or four copies by ordinary mail. I am a member of the steering committee of the Advances in Modal Logic Initiative (AiML) I am a member of the organising committee of the annual Automated Reasoning Workshop Involvement with scientific meetings as a member of the program committee: Advances in Modal Logics 2004 , 9-11 September 2004, Manchester (local organiser) European Conference on Logics in Artificial Intelligence , 27-30 September 2004, Lisbon Computational Logic in Multi-Agent Systems V, 28-29 September 2004, Lisbon

93. ABSTRACT ALGEBRA ON LINE Contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. Intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. http://www.math.niu.edu/~beachy/aaol/

WELCOME TO ABSTRACT ALGEBRA ON LINE This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the books Abstract Algebra , by John A. Beachy and William D. Blair, and Abstract Algebra II , by John A. Beachy. The site is organized by chapter. The page containing the Table of Contents also contains an index of definitions and theorems, which can be searched for detailed references on subject area pages. Topics from the first volume are marked by the symbol and those from the second volume by the symbol . To make use of this site as a reference, please continue on to the Table of Contents. TABLE OF CONTENTS (No frames) TABLE OF CONTENTS (Frames version) Interested students may also wish to refer to a closely related site that includes solved problems: the OnLine Study Guide for Abstract Algebra REFERENCES Abstract Algebra Second Edition , by John A. Beachy and William D. Blair

94. Boolean Algebra normal algebra (table 7.3). Table 7.3 Boolean commutative, distributive and associative rules. We will also make extensive use of De Morgan s theorems (table http://www.phys.ualberta.ca/~gingrich/phys395/notes/node121.html

Next: Logic Gates Up: Digital Circuits Previous: Number Representation Boolean Algebra The binary and 1 states are naturally related to the true and false logic variables. We will find the following Boolean algebra useful. Consider two logic variables A and B and the result of some Boolean logic operation Q . We can define Q is true if and only if A is true AND B is true. Q is true if A is true OR B is true. Q is true if A is false. A useful way of displaying the results of a Boolean operation is with a truth table. We will make extensive use of truth tables later. If no ``-'' is available on your text processor or circuit drawing program an `` N '' can be used, ie. We list a few trivial Boolean rules in table Table 7.2: Properties of Boolean Operations. The Boolean operations obey the usual commutative, distributive and associative rules of normal algebra (table Table 7.3: Boolean commutative, distributive and associative rules. We will also make extensive use of De Morgan's theorems (table Table 7.4: De Morgan's theorems. Doug Gingrich Tue Jul 13 16:55:15 EDT 1999

95. DeMorgan's Theorems - Chapter 7: BOOLEAN ALGEBRA - Volume IV - Digital gate Now, we reduce this expression using the identities, properties, rules, and theorems (DeMorgan s) of Boolean algebra The http://www.allaboutcircuits.com/vol_4/chpt_7/8.html

Volume I - DC Volume II - AC Volume III - Semiconductors Volume IV - Digital ... Back to Chapter Index Search All Volumes Volume I - DC Volume II - AC Volume III - Semiconductors Volume IV - Digital Volume V - Reference Volume VI - Experiments Check out our new Electronics Forums Ask questions and help answer others. Check it out! DeMorgan's Theorems All About Circuits Volume IV - Digital Chapter 7: BOOLEAN ALGEBRA DeMorgan's Theorems DeMorgan's Theorems A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra. By group complementation, I'm referring to the complement of a group of terms, represented by a long bar over more than one variable. You should recall from the chapter on logic gates that inverting all inputs to a gate reverses that gate's essential function from AND to OR, or visa-versa, and also inverts the output. So, an OR gate with all inputs inverted (a Negative-OR gate) behaves the same as a NAND gate, and an AND gate with all inputs inverted (a Negative-AND gate) behaves the same as a NOR gate. DeMorgan's theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs: A long bar extending over the term AB acts as a grouping symbol, and as such is entirely different from the product of A and B independently inverted. In other words, (AB)' is not equal to A'B'. Because the "prime" symbol (') cannot be stretched over two variables like a bar can, we are forced to use parentheses to make it apply to the whole term AB in the previous sentence. A bar, however, acts as its own grouping symbol when stretched over more than one variable. This has profound impact on how Boolean expressions are evaluated and reduced, as we shall see.

96. ABSTRACT ALGEBRA ON LINE: Structure Of Groups Excerpted from Beachy/Blair, Abstract algebra, 2nd Ed., © 1996 Chapter 7. 7.1 Isomorphism theorems; automorphisms 7.2 Conjugacy 7.3 Groups acting on sets 7.4 http://www.math.niu.edu/~beachy/aaol/structure.html

STRUCTURE OF GROUPS Excerpted from Beachy/Blair, Abstract Algebra 2nd Ed. Chapter 7 Isomorphism theorems; automorphisms Conjugacy Groups acting on sets The Sylow theorems ... Simple groups Excerpted from Abstract Algebra: Supplementary Lecture Notes Nilpotent groups Semidirect products Classification of groups of small order ... About this document Isomorphism theorems; automorphisms 7.1.1. Theorem. [First Isomorphism Theorem] Let G be a group with normal subgroups N and H such that N H. Then H/N is a normal subgroup of G/N, and (G / N) / (H / N) G / H. 7.1.2. Theorem. [Second Isomorphism Theorem] Let G be a group, let N be a normal subgroup of G, and let H be any subgroup of G. Then HN is a subgroup of G, H N is a normal subgroup of H, and (HN) / N H / (H N). 7.1.3. Theorem. Let G be a group with normal subgroups H, K such that HK=G and H G H K . 7.1.4. Proposition. Let G be a group and let a G. The function i a a (x) = axa for all x G is an isomorphism. 7.1.5. Definition. Let G be a group. An isomorphism from G onto G is called an automorphism of G. An automorphism of G of the form i

97. Chapter 4 Boolean Algebra algebra. 41 Describing Logic Circuits algebraically. 4-2 Evaluating Logic Circuit Outputs. 4-3 Implementing Circuits from Boolean Expression. 4-4 Boolean Theorems. http://www.eelab.usyd.edu.au/digital_tutorial/chapter4/4_0.html

Chapter 4 Boolean Algebra 4-1 Describing Logic Circuits Algebraically 4-2 Evaluating Logic Circuit Outputs 4-3 Implementing Circuits from Boolean Expression 4-4 Boolean Theorems 4-5 DeMorgan's Theorems 4-6 Universality of NAND and NOR Gates 4-7 Alternate Logic-Gate Representations 4-8 Logic Symbol Interpretation Let's Go to QUIZ 4

98. Morera's Theorem. Previous Generalization of Cauchy s Integral Contents Morera s theorem. Using this result we can prove the following converse to CauchyGoursat theorem http://ndp.jct.ac.il/tutorials/complex/node37.html

Next: The theorems of Liouville Up: Integrals. Previous: Generalization of Cauchy's Integral Contents Morera's Theorem. In two variable Calculus, you learnt the following result: let and be two functions defined on the same simply connected domain . Suppose that and are continuous on the interior of and that, for every Jordan curve in , the following equation holds: Then in Using this result we can prove the following converse to Cauchy-Goursat theorem: Proposition 5.6.1 Let be a function such that are continuous in a simply connected domain . Suppose that, for every Jordan curve in , the integral is equal to 0. Then is analytic on Another converse of Cauchy-Goursat theorem, stronger than Prop. is Morera's theorem: Theorem 5.6.2 (Morera) Let be a continuous function on an open simply connected domain . Assume that for every loop in , the integral is equal to 0. Then is analytic on Example 5.6.3 Next: The theorems of Liouville Up: Integrals. Previous: Generalization of Cauchy's Integral Contents Noah Dana-Picard 2004-01-26

99. Pythagoras' Theorem - Dissections Conceptually, one of the most attractive ways to prove Pythagoras theorem is to them turn out to be closely related to the straightforward `algebraic proofs http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagorasdissecti

Proofs of Pythagoras' Theorem using translations Conceptually, one of the most attractive ways to' prove Pythagoras' Theorem is to find a partition the three squares into smaller regions, with the property that the partition of the large square is assembled from regions in the partitions of the two other squares. Such proofs are among the oldest known. Some of them turn out to be closely related to the straightforward `algebraic' proofs. There is in fact a general and elementary result about areas in the plane (apparently proven first around 1900) which asserts that if two polygonal figures in the plane have the same area, then one can find piecewise congruent decomposition of the two figures. That is to say, on grounds of general principle one knows that if Pythagoras' Theorem is true, one can find the sort of partitions we are looking for. Reference Heinz Hopf, Springer Lecture Notes in Mathematics #1000. David Hilbert, Foundations of Geometry , Open House, 1994. Proof by algebra This has apparently been found independently many times. On each side we have a square of side a + b . On the left inscribed in it is a square of side c , and four copies of the original right triangle. The area of the square on the left is therefore

100. Direct Consequences next up previous contents index Next Sums, Products and Quotients Up Sequences Previous Examples of sequences Contents Index Direct Consequences. http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node19.html

Next: Sums, Products and Quotients Up: Sequences Previous: Examples of sequences Contents Index Direct Consequences With this language we can give some simple examples for which we can use the definition directly. If a n 2 as n , then (take = 1), eventually, a n is within a distance 1 of 2. One consequence of this is that eventually a n a n Let a n n . Then a n as n . To check this, pick N with N . Now suppose that n N . We have by choice of N The sequence a n n - 1 is divergent; for if not, then there is some l such that a n l as n . Taking = 1, we see that eventually (say after N ) , we have n l n l n N . thus n l + 2 for all n , which is a contradiction. Exercise 2.5 Show that the sequence a n as n Although we can work directly from the definition in these simple cases, most of the time it is too hard. So rather than always working directly, we also use the definition to prove some general tools, and then use the tools to tell us about convergence or divergence. Here is a simple tool (or Proposition). Proposition 2.6 Let a n l as n and assume also that a n m as n . Then l m . In other words, if a sequence has a limit, it has a unique limit, and we are justified in talking about

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