1.4-5 Theorem (Convergent sequence). Every convergent sequence in a metric space is a Cauchy sequence. that = Proof. If x →→→→→→x, then for every 8>0 there is an N-N(s) such xn d(x, x)< Hence by the triangle inequality we obtain for m, n>N E E d(xm, xn)≤d(xm, x)+d(x, xn)<±±±²² = 8. This shows that (x) is Cauchy. for all > 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 8: Complex Convergence in Product Metric Spaces with Variable Components Problem Statement: Let X = C([0,1], R) × (², where: C([0, 1], R) is the space of continuous real-valued functions on the interval [0,1] with the supremum norm ||f||∞ = supr¤[0,1] |ƒ(x)|. l² is the space of square-summable real sequences with the standard ² norm ||3y||2 = [鰯) 1/2. Equip X with the product metric d defined by: d((fi, y¹), (f2, y²)) = || f1 - f2|| + ||y¹ - y²||2. Consider the sequence {*} in X where each z = (fk, y) is defined by: f(x) = x, y 1 1 k' k 1,0,0,...) with the first & terms equal to 1. a. Analyze the convergence of the sequence {f} in C([0, 1], R) with respect to the supremum norm. Identify the limit function if convergence occurs. 2. b. Examine the convergence of the sequence {*} in 12. Determine whether {*} converges and identify the limit if it exists. 3. c. Determine whether the sequence {} converges in X with the product metric d. Provide a detailed justification based on the convergence of its components.

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