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- Let x1 > 1 and xn+1 = 2 − 1 / xn for n ≥ 2. Show that ( xn ) converges and find its limit.In Example 2.4.1, show that fn(x) converges to f(x) pointwise on [0, 1]. (We haveseen the convergence at x = 0, 1.) For x ∈ [0, 1] and positive integer n, let fn(x) = xn.Then limn→∞ fn(x) = f(x),, wheref(x) = 0 if 0 ≤ x < 1,1 if x = 1.This example shows that the pointwise limit of a sequence of continuous functions need not be continuous.Let x_1 = 1/2 and, for n ≥ 1, x_(n+1) = √xn. Prove that the sequence (xn)^∞_n=1 converges and find its limit.
- (b) A sequence (fn) of differentiable functions such that (fn ) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.1. Consider the sequence Xn = √n + 1 − √n, n ≥ 1. Prove that (xn)n isconvergent. Find its limit.Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for k = 1 to infinity converges and find its limit.
- (a) Why does the sequence of real numbers fn(x1) necessarily contain a convergent subsequence (fnk )? To indicate that the subsequence of functions (fnk ) is generated by considering the values of the functions at x1, we will use the notation fnk = f1,k.Determine whether the sequence xn =√(5+3n+7n2)/(4n2-n+2), converges or diverges. If it converges, find the limitSolve for A and B so that F(x) has a limit at both x=2 and x=4