1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. Proof. If xx, then for every & >0 there is an N = N(&) such that d(xn, x)< for all n > N. do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig. Problem 5: Compactness and Convergence in Function Spaces with Weighted Metrics Problem Statement: Consider the space C(R) of all bounded continuous real-valued functions on R, equipped with the weighted supremum metric Hence by the triangle inequality we obtain for m, n>N <+들 8. d(xm, xn)≤d(xm, x)+d(x, xn).
1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. Proof. If xx, then for every & >0 there is an N = N(&) such that d(xn, x)< for all n > N. do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig. Problem 5: Compactness and Convergence in Function Spaces with Weighted Metrics Problem Statement: Consider the space C(R) of all bounded continuous real-valued functions on R, equipped with the weighted supremum metric Hence by the triangle inequality we obtain for m, n>N <+들 8. d(xm, xn)≤d(xm, x)+d(x, xn).
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 25E
Question
![1.4-5 Theorem (Convergent sequence). Every convergent sequence in
a metric space is a Cauchy sequence.
Proof. If xx, then for every & >0 there is an N = N(&) such
that
d(xn, x)<
for all n > N.
do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig.
Problem 5: Compactness and Convergence in Function Spaces with
Weighted Metrics
Problem Statement:
Consider the space C(R) of all bounded continuous real-valued functions on R, equipped with the
weighted supremum metric
Hence by the triangle inequality we obtain for m, n>N
<+들
8.
d(xm, xn)≤d(xm, x)+d(x, xn)</
This shows that (x,) is Cauchy.
We shall see that quite a number of basic results, for instance in
the theory of linear operators, will depend on the completeness of the
corresponding spaces. Completeness of the real line R is also the main
reason why in calculus we use R rather than the rational line Q (the set
of all rational numbers with the metric induced from R).
Let us continue and finish this section with three theorems that are
related to convergence and completeness and will be needed later.
d(9) sup
ZER
|f(x) − g(x)|
1+2
Let {f} be a sequence of functions defined by:
sin(nz)
$(2)=
1+2
1. a. Prove whether {f} is a Cauchy sequence in C() with respect to the weighted supremum
metric d
2. b. Determine if {f} converges in C, (R) under d. If it does, identify the limit function. If not,
provide a detailed explanation.
3. c. Investigate the pointwise convergence of {f} on R. Does pointwise convergence imply
convergence in C (R) with the given metric?
4. d. Visualize the sequence {f} by plotting fn() for n = 1,5, 10, 20 over the interval z
[-10, 10]. Analyze how these plots reflect the behavior of convergence in both pointwise and
metric senses.
5. e. Explore the compactness of the set {f} in C(R) with d.. Utilize the Arzela-Ascoli Theorem
by verifying the necessary conditions (equicontinuity and pointwise boundedness) and discuss
whether {f} possesses a convergent subsequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34f1dca3-855f-4ba4-82f8-d2c60fd46264%2F88dec668-2d82-45ad-9fa8-8dbc9ff05d53%2Fxmfdpyh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.4-5 Theorem (Convergent sequence). Every convergent sequence in
a metric space is a Cauchy sequence.
Proof. If xx, then for every & >0 there is an N = N(&) such
that
d(xn, x)<
for all n > N.
do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig.
Problem 5: Compactness and Convergence in Function Spaces with
Weighted Metrics
Problem Statement:
Consider the space C(R) of all bounded continuous real-valued functions on R, equipped with the
weighted supremum metric
Hence by the triangle inequality we obtain for m, n>N
<+들
8.
d(xm, xn)≤d(xm, x)+d(x, xn)</
This shows that (x,) is Cauchy.
We shall see that quite a number of basic results, for instance in
the theory of linear operators, will depend on the completeness of the
corresponding spaces. Completeness of the real line R is also the main
reason why in calculus we use R rather than the rational line Q (the set
of all rational numbers with the metric induced from R).
Let us continue and finish this section with three theorems that are
related to convergence and completeness and will be needed later.
d(9) sup
ZER
|f(x) − g(x)|
1+2
Let {f} be a sequence of functions defined by:
sin(nz)
$(2)=
1+2
1. a. Prove whether {f} is a Cauchy sequence in C() with respect to the weighted supremum
metric d
2. b. Determine if {f} converges in C, (R) under d. If it does, identify the limit function. If not,
provide a detailed explanation.
3. c. Investigate the pointwise convergence of {f} on R. Does pointwise convergence imply
convergence in C (R) with the given metric?
4. d. Visualize the sequence {f} by plotting fn() for n = 1,5, 10, 20 over the interval z
[-10, 10]. Analyze how these plots reflect the behavior of convergence in both pointwise and
metric senses.
5. e. Explore the compactness of the set {f} in C(R) with d.. Utilize the Arzela-Ascoli Theorem
by verifying the necessary conditions (equicontinuity and pointwise boundedness) and discuss
whether {f} possesses a convergent subsequence.
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