The Least Upper Bound Axiom AXIOM 11.1.1 THE LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. Example: (1)lub(-o,0)=0,lub(-oo,01=0 (2)lub(-4,-1)=-1,lub(-4,-1]=-1 (3)lub{1/2,2/3,3/4,..,n/(n+1),..}= (4)lub{-1/2,-1/8,-1/27,.,-1/m3,..}= [Main Menu☐ cmew5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom Example: (1) lub (−∞, 0) = 0, lub(−∞, 0] = 0 (2) lub (−4,−1) = −1, lub(−4,−1] = -1 (3) lub {1/2, 2/3, 3/4, . . . , n/(n + 1), . . . } = (4) lub {−1/2,−1/8,−1/27, . . . ,−1/n3 , . . . } =
(⑤)lub{x:x2<3}=lub{x:-5<x<√3} (6)For each decimal fraction b=0.bbb3.... we have b=lub (0.b1.0.bb2,0.bb2b3....). (7)If S consists of the lengths of all polygonal paths inscribed in a semicircle of radius 1,then lub S=(halfthe circumference of the unit circle). Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. (5) lub {x : x 2 < 3} = lub{x : } (6) For each decimal fraction b = 0.b1b2b3 , . . . , we have b = lub {0.b1 , 0.b1b2 , 0.b1b2b3 , . . . }. (7) If S consists of the lengths of all polygonal paths inscribed in a semicircle of radius 1, then lub S = π (half the circumference of the unit circle). − 3 3 x
The Least Upper Bound Axiom THEOREM 11.1.2 If M is the least upper bound of the set S and e is a positive number,then there is at least one number s in S such that M-e<s≤M. Main Meny o8的大
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom
The Least Upper Bound Axiom Example (a)LetS={1/2,2/3,3/4,..,n/n+1),..}and take s=0.0001. (b)LetS=(0.1,2.3)and take s=0.00001. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom Example (a) Let S = {1/2, 2/3, 3/4, . . . , n/(n + 1), . . . } and take ε = 0.0001. (b) Let S = {0, 1, 2, 3} and take ε = 0.00001
The Least Upper Bound Axiom THEOREM 11.1.3 Every nonempty set of real numbers that has a lower bound has a greatest lower bound. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom
The Least Upper Bound Axiom THEOREM 11.1.4 If m is the greatest lower bound of the set S and e is a positive number,then there is at least one number s in S such that m≤s<m+e. Main Meny o
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Sequences of Real Numbers DEFINITION 11.2.1 SEQUENCE OF REAL NUMBERS A sequence of real numbers is a real-valued function defined on the set of positive integers. Main Menu cmew5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers
Sequences of Real Numbers For example: g,=1 is the sequence 29 n b=n is the sequence 1234 n+1 2’3?435? c n2 is the sequence 1,4,9,16,... Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers For example: 2 111 234 1 2 3 4 2 3 4 5 1 is the sequence 1, , , , . . . is the sequence , , , , . . . 1 is the sequence 1, 4, 9, 16, . . . n n n a n n b n c n = = + =
Sequences of Real Numbers From a az as....am...and b b2 b3.....bm... we can form the scalar product sequence aa,adz,aa3,...,aa,..., the sum sequence a+b,az+bz,a3+b3,...,an+b..... the difference sequence a-b,a2-b2,a3-b3....,an-bn...., the product sequence:ab,azb2,a3b3,....ab..... If b;0 for all i,we can form the reciprocal sequence .1111 6’6’6’五’ a az as a the quotient sequence Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers From a1 , a2 , a3 , . . . , an , . . . and b1 , b2 , b3 , . . . , bn , . . . we can form the scalar product sequence : αa1 , αa2 , αa3 , . . . , αan , . . . , the sum sequence : a1 + b1 , a2 + b2 , a3 + b3 , . . . , an + bn , . . . , the difference sequence : a1 − b1 , a2 − b2 , a3 − b3 , . . . , an − bn , . . . , the product sequence : a1b1 , a2b2 , a3b3 , . . . , anbn , . . . . If bi ≠ 0 for all i , we can form the quotient sequence : the reciprocalsequence : 1 2 3 1 1 1 1 , , , . . . , , . . . n b b b b 1 2 3 1 2 3 , , , . . . , , . . . n n a a a a b b b b
Sequences of Real Numbers The sequence with terms a,is said to be increasing if an<a for all n, nondecreasing if anam for all n. decreasing if an aml for all n, nonincreasing if an≥for all n. A sequence that satisfies any of these conditions is called monotonic. Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers The sequence with terms an is said to be increasing if an an+1 for all n, nonincreasing if an ≥ an+1 for all n. A sequence that satisfies any of these conditions is called monotonic