1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. that Proof. If xx, then for every ɛ>0 there is an N = N(E) such 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 4: Sequential Convergence in Infinite-Dimensional Banach Spaces Problem Statement: Let denote the space of all bounded real sequences, equipped with the supremum norm ||||∞ = supenn. Consider the following sequence of sequences {a} in, where each is defined by: Hence by the triangle inequality we obtain for m, n>N E Ε 22 d(xm, xn)≤d(xm, x)+d(x, xn) <=+= = 8. This shows that (x,) is Cauchy. I 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. if n≤k, if n > k. 1. a. Prove or disprove that the sequence {a} converges in l. If it converges, identify the limit. If not, explain why convergence fails. 2. b. Analyze the convergence of {*} in the weak* topology of *. Determine whether {z} converges under this topology and identify the limit if it exists. 3. c. Visualize the behavior of {"} by plotting the first 20 terms of each sequence for k = 1,5, 10, 15, 20. Discuss how these plots illustrate the convergence or lack thereof in both the norm and weak* topologies. 4. d. Extend the analysis to the space co (the space of sequences converging to zero) with the supremum norm. Determine whether {z} converges in co and provide graphical evidence to support your conclusion.

1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. do by hand, without AI, I need detailed, graphs and codes also, make sure to answer using kresjig. Proof. If xn →x, then for every & >0 there is an N = N(&) such that d(x, x) N Hence by the triangle inequality we obtain for m, n>N E E d(xm, xn)≤d(xm, x)+d(x, xn) < ±2+2 = 8. 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. Problem 1: Convergence in Function Spaces with Supremum Metric Problem Statement: Let C([0,1]) denote the space of…