1.1. Let f: [a, b] → [a, b] be such that f(x1)-f(xo)| ≤ L1 - xol for all o, 21 € [a, b], with L < 1. Show there is a [a, b] such that f(x) = <= x. Define a sequence by induction as follows. Let zo E€ [a, b] be arbitrary. Once -1 € [a, b] has been defined, let n = f(n-1). Show that (n)o is a Cauchy sequence, then use that R is complete and that [a, b] is closed. Then show that if z limnoo Zn, then = f(x).
1.1. Let f: [a, b] → [a, b] be such that f(x1)-f(xo)| ≤ L1 - xol for all o, 21 € [a, b], with L < 1. Show there is a [a, b] such that f(x) = <= x. Define a sequence by induction as follows. Let zo E€ [a, b] be arbitrary. Once -1 € [a, b] has been defined, let n = f(n-1). Show that (n)o is a Cauchy sequence, then use that R is complete and that [a, b] is closed. Then show that if z limnoo Zn, then = f(x).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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![1.1. Let f [a, b] → [a, b] be such that f(x₁)-f(xo)| ≤ L1 - col for all to, 1 €
[a, b], with L < 1. Show there is a [a, b] such that f(x) =
<= x.
Define a sequence by induction as follows. Let zo E€ [a, b] be arbitrary. Once -1 € [a, b]
has been defined, let n = f(n-1). Show that (an)o is a Cauchy sequence, then use that
R is complete and that [a, b] is closed. Then show that if a limnoon, then = f(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34b44b37-e66b-4c48-a846-aeedd1db2b78%2Fc375cc87-5c47-40f1-82f2-ac5a07cc3b8e%2F2dq8fcj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.1. Let f [a, b] → [a, b] be such that f(x₁)-f(xo)| ≤ L1 - col for all to, 1 €
[a, b], with L < 1. Show there is a [a, b] such that f(x) =
<= x.
Define a sequence by induction as follows. Let zo E€ [a, b] be arbitrary. Once -1 € [a, b]
has been defined, let n = f(n-1). Show that (an)o is a Cauchy sequence, then use that
R is complete and that [a, b] is closed. Then show that if a limnoon, then = f(x).
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