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)< for all n> 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 continuous real-valued functions on the interval [0, 1] equipped with the supremum metric d (f,g) = supɛɛ|0,1] |ƒ(z) − g(x)|. Consider the sequence of functions {n} defined by: fn(x) = x² 1. a. Prove that {f} converges in C([0, 1]) with respect to the supremum metric. If it converges, identify the limit function f. 2. b. Analyze the uniform convergence of {f} on [0, 1]. Is the convergence uniform? Justify your answer using graphical intuition. 3. c. Now, consider the sequence {n} where: 9n(x) = Determine whether {g} converges in C([0, 1]) with respect to the supremum metric. Provide a proof and illustrate your reasoning with appropriate graphs.