The Cauchy Criterion

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1 The Cauchy Criterion MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007

2 Cauchy Sequences Definition A sequence X = (x n ) of real numbers is a Cauchy sequence if it satisfies the Cauchy criterion: for all ǫ > 0 there exists H(ǫ) N such that for all m, n H(ǫ), then terms x m, x n satisfy x m x n < ǫ. Example ( ) 1 is a Cauchy sequence. n (1 + ( 1) n ) is not a Cauchy sequence.

3 Cauchy Sequences Definition A sequence X = (x n ) of real numbers is a Cauchy sequence if it satisfies the Cauchy criterion: for all ǫ > 0 there exists H(ǫ) N such that for all m, n H(ǫ), then terms x m, x n satisfy x m x n < ǫ. Example ( ) 1 is a Cauchy sequence. n (1 + ( 1) n ) is not a Cauchy sequence.

4 Cauchy Sequences Definition A sequence X = (x n ) of real numbers is a Cauchy sequence if it satisfies the Cauchy criterion: for all ǫ > 0 there exists H(ǫ) N such that for all m, n H(ǫ), then terms x m, x n satisfy x m x n < ǫ. Example ( ) 1 is a Cauchy sequence. n (1 + ( 1) n ) is not a Cauchy sequence.

5 Cauchy vs. Convergent Lemma If X = (x n ) is a convergent sequence of real numbers, then X is a Cauchy sequence. Lemma A Cauchy sequence of real numbers is bounded.

6 Cauchy vs. Convergent Lemma If X = (x n ) is a convergent sequence of real numbers, then X is a Cauchy sequence. Lemma A Cauchy sequence of real numbers is bounded.

7 Cauchy vs. Convergent Lemma If X = (x n ) is a convergent sequence of real numbers, then X is a Cauchy sequence. Lemma A Cauchy sequence of real numbers is bounded.

8 Cauchy vs. Convergent Lemma If X = (x n ) is a convergent sequence of real numbers, then X is a Cauchy sequence. Lemma A Cauchy sequence of real numbers is bounded.

9 Cauchy Convergence Criterion Theorem (Cauchy Convergence Criterion) A sequence of real numbers is convergent if and only if it is a Cauchy sequence.

10 Cauchy Convergence Criterion Theorem (Cauchy Convergence Criterion) A sequence of real numbers is convergent if and only if it is a Cauchy sequence.

11 Examples Example Consider the sequence: x 1 = 1, x 2 = 2, and x n = 1 2 (x n 2 + x n 1 ) for n > 2. Consider the sequence y 1 = 1 1!, y 2 = 1 1! 1 2!,, y n = 1 1! 1 2! Consider the sequence + +( 1)n+1, n! z 1 = 1 1, z 2 = ,, z n = n,

12 Examples Example Consider the sequence: x 1 = 1, x 2 = 2, and x n = 1 2 (x n 2 + x n 1 ) for n > 2. Consider the sequence y 1 = 1 1!, y 2 = 1 1! 1 2!,, y n = 1 1! 1 2! Consider the sequence + +( 1)n+1, n! z 1 = 1 1, z 2 = ,, z n = n,

13 Examples Example Consider the sequence: x 1 = 1, x 2 = 2, and x n = 1 2 (x n 2 + x n 1 ) for n > 2. Consider the sequence y 1 = 1 1!, y 2 = 1 1! 1 2!,, y n = 1 1! 1 2! Consider the sequence + +( 1)n+1, n! z 1 = 1 1, z 2 = ,, z n = n,

14 Contractive Sequences Definition A sequence of real numbers X = (x n ) is contractive if there exists a constant 0 < C < 1 such that for all n N. x n+2 x n+1 C x n+1 x n Theorem Every contractive sequence is a Cauchy sequence and therefore is convergent.

15 Contractive Sequences Definition A sequence of real numbers X = (x n ) is contractive if there exists a constant 0 < C < 1 such that for all n N. x n+2 x n+1 C x n+1 x n Theorem Every contractive sequence is a Cauchy sequence and therefore is convergent.

16 Contractive Sequences Definition A sequence of real numbers X = (x n ) is contractive if there exists a constant 0 < C < 1 such that for all n N. x n+2 x n+1 C x n+1 x n Theorem Every contractive sequence is a Cauchy sequence and therefore is convergent.

17 Further Results Corollary If X = (x n ) is a contractive sequence with constant 0 < C < 1 and if lim X = L, then 1 x n L Cn 1 1 C x 2 x 1 2 x n L C 1 C x n x n 1

18 Further Results Corollary If X = (x n ) is a contractive sequence with constant 0 < C < 1 and if lim X = L, then 1 x n L Cn 1 1 C x 2 x 1 2 x n L C 1 C x n x n 1

19 Application Example Approximate the solution 0 < x < 1 to the equation x 3 7x + 2 = 0

20 Homework Read Section 3.5. Page 86: 1, 2, 3, 7, 8, 9, 11, 14 Boxed problems should be written up separately and submitted for grading at class time on Friday.

Monotone Sequences & Cauchy Sequences Philippe B. Laval

Monotone Sequences & Cauchy Sequences Philippe B. Laval Monotone Sequences & Cauchy Sequences 2 1 Monotone Sequences and Cauchy Sequences 1.1 Monotone Sequences The techniques we have studied so far require