1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. that Proof. If xn →x, then for every & >0 there is an N = N(E) such d(xx)< Hence by the triangle inequality we obtain for m, n>N E d(xx) = d(x, x)+d(x, x) << ±±± This shows that (x) is Cauchy. I for all n > N. 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. do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig. Problem 2: Compactness and Sequential Convergence in Metric Spaces Problem Statement: Let (X,d) be a metric space where X = {(x, y) = R² | ² + y² ≤1} (the closed unit disk) with the standard Euclidean metric. 1. a. Prove that every sequence in X has a convergent subsequence whose limit lies in X (i.e., X is sequentially compact). 2. b. Construct a specific sequence {P} in X such that: p = (cos (2), sin (2)) for some fixed integer : > 3. Determine whether {P} has a convergent subsequence in X. If so, describe all possible limits. 3. c. Visualize the behavior of sequences in X by graphing three different sequences: ⚫ One converging to a point inside the disk. • One converging to a boundary point. One that does not converge but has convergent subsequences. Provide a detailed explanation of each case based on the graphical representations.
1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. that Proof. If xn →x, then for every & >0 there is an N = N(E) such d(xx)< Hence by the triangle inequality we obtain for m, n>N E d(xx) = d(x, x)+d(x, x) << ±±± This shows that (x) is Cauchy. I for all n > N. 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. do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig. Problem 2: Compactness and Sequential Convergence in Metric Spaces Problem Statement: Let (X,d) be a metric space where X = {(x, y) = R² | ² + y² ≤1} (the closed unit disk) with the standard Euclidean metric. 1. a. Prove that every sequence in X has a convergent subsequence whose limit lies in X (i.e., X is sequentially compact). 2. b. Construct a specific sequence {P} in X such that: p = (cos (2), sin (2)) for some fixed integer : > 3. Determine whether {P} has a convergent subsequence in X. If so, describe all possible limits. 3. c. Visualize the behavior of sequences in X by graphing three different sequences: ⚫ One converging to a point inside the disk. • One converging to a boundary point. One that does not converge but has convergent subsequences. Provide a detailed explanation of each case based on the graphical representations.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 64E
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Transcribed Image Text:1.4-5 Theorem (Convergent sequence). Every convergent sequence in
a metric space is a Cauchy sequence.
that
Proof. If xn →x, then for every & >0 there is an N = N(E) such
d(xx)<
Hence by the triangle inequality we obtain for m, n>N
E
d(xx) = d(x, x)+d(x, x) << ±±±
This shows that (x) is Cauchy. I
for all n > N.
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.
do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig.
Problem 2: Compactness and Sequential Convergence in Metric Spaces
Problem Statement:
Let (X,d) be a metric space where X = {(x, y) = R² | ² + y² ≤1} (the closed unit disk) with
the standard Euclidean metric.
1. a. Prove that every sequence in X has a convergent subsequence whose limit lies in X (i.e., X
is sequentially compact).
2. b. Construct a specific sequence {P} in X such that: p = (cos (2), sin (2)) for some
fixed integer : > 3. Determine whether {P} has a convergent subsequence in X. If so,
describe all possible limits.
3. c. Visualize the behavior of sequences in X by graphing three different sequences:
⚫ One converging to a point inside the disk.
• One converging to a boundary point.
One that does not converge but has convergent subsequences.
Provide a detailed explanation of each case based on the graphical representations.
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