26.15. (a) Show that if () is tight, then the characteristic functions (1) are uniformly equicontinuous (for each e there is a & such that s-t<8 implies that (s)-(1)| <€ for all n). (b) Show that μμ implies that (1)→ (1) uniformly on bounded sets. n (c) Show that the convergence in part (b) need not be uniform over the entire line.

Q: fk (x) = fk (x) 9k (x) = { 9k (x) = = = sin x k X 1+kx² if |x ≤ k if |x|> k. ((-1²) c -) cos(kx)

A:

Q: 4. Consider the sequence of functions : f₁(x) = 1 n³ [x−²+ +1 (a) the uniform topology (If yes, to…

A: The given sequence of function is ;    fn(x) = 1n3x-1n2+1 We have to answer the following questions…

Q: S1, S2, 2. Show that if (fn)n converges uniformly on U- Sk. S,, then (fn)n converges uniformly on…

A:

Q: .3 Let fn : [-3,3] → R be defined by fn(x) = . Find the pointwise limit of fn(x). 2n+1' .4 Does…

A: Given,         fn : [-3 ,3] -&gt; R be defined by ,                             fn(x) = x(2n)/{x(2n)…

Q: S² P, o² and „", → 1, their product converges in *6. In lecture it is claimed that since "S2…

A: Given: Yn is a sequence of random variables that converges in probability to Y. So,…

Q: 6. Let G = [0, 2] and 1 g(x) -r + 4). 4 3 3 Use the contraction mapping theorem to prove that if ro…

A: given G=0,2 and  gx=13x33-x2-54x+4

Q: 1.1 Let {fn(x)} = } , x € R and r > 0. + xn 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A:

Q: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

A:

Q: 2. Let {fn(x)} be a sequence of real-valued functions on [a, b]. Suppose that {fn} is uniformly…

A:

Q: Assume that a > 0. For every n E N* and a E (0, 00, let fn(x) = n"re-n. (a) Show that the sequence…

A: We are given that for every n∈ℕ*and x∈[ 0 , ∞)  ,  fn (x) = nαx e-nx  , where  α&gt;0  To show :…

Q: et inneN be a sequence of continuous functions on some set S that converges uniformly to f. Let…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: 2. Consider the following sequence of closed subintervals of [0, 1]: | 0, 2 1 3 3 and denote the nth…

A: The given function is fn(x)=1  for x in In,0  for x not in In. In denote the nth subinterval. The…

Q: 3. Find the pointwise limit of the sequence of functions defined on R by the following and determine…

A: To find the pointwise limit of the sequence of functions and determine whether convergence is…

Q: 1.1 Let {fn (2)} = {1 NI +nz ,z ER and 2 > 0. 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A: To Examine: The series of function Sn(x)=∑k=1∞xk-1 converges uniformly on [-a,a], a∈(0,1) or not.

Q: functions such that f→fa.e. on a measurable set E of finite measure, then given n> 0 there is a…

A:

Q: functions such that f→fa.e. on a measurable set E of finite measure, then given >0 there is a subset…

A:

Q: (d) 5 (ar + 2)k 2k k=1

A:

Q: 1.1 Let {fn(x)} = { ,x ER and r > 0. %3D + xn 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A:

Q: 10. Find the radius of convergence of the following: k!æk (a) (b) (k!)?pt (2k)! k=1

A:

Q: From first principles (that is, using the formal definitions of convergence and divergence) show…

A:

Q: Prae that x4 -5x +3 is ineducible ouer Q

A: Given polynomial is f(x)= x^4 -5x^3 +3

Q: 3. For each of the following sequences of real-valued functions on R, use the definition to show…

A: Answer

Q: 5. Let P([0, 1]) be the space of polynomials on [0, 1] with the norm of uniform convergence, that…

A:

Q: 1.1 Show that {fn(x)} converges uniformly on [a, b] if and only if given any E > 0 there is a…

A:

Q: (4.3) Let {X;}1<i<k be a collection of complete metric spaces. (a) Show that every Cauchy sequence…

A: A metric space (X, d) is complete if every Cauchy sequence of points in X has a limit that is also…

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n²x; if x E (0, ) = (x)"f 0; otherwise Find the…

A: we have sequence of function fnx=n-n2x ; if x∈0,1n0;           otherwise pointwise limit of sequence…

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

A:

Q: 2.69 Let if 0 < x <, n fn (x) = 0 if x € (0, 1) \ (0, 4), for all n E N. For each a E [0, 1], find…

A:

Q: Let f(r) = 15x + 6a? – . Can Golden Ratio search method be used to find a local minimum of f…

A:

Q: Q5. Given f(x): (R.) → (R₁) 3x+2 if x 22 5x-1 if x < 2 f(x) = { (a) Let {n} = {2} be a sequence in…

A:

Q: Argue that Given a continuous function g defined on [0, 1] define functions G₂(2) = 9 (-) (*)2²…

A: It is well established that almost sure convergence of a sequence of random variables Yn→a.sc…

Q: 20.22. (a) Suppose that X₁ ≤X₂ ≤. ... and that Xp X. Show that X, →X with n probability 1. (b) Show…

A: From the above details we have.

Q: Let (un) and (Vn) be two sequences defined by n + "n Un+1 Un + Vn 0 < v1 < U1, Un+1 Un + Vn 2 (a)…

A: Given un and vn are two sequences defined by 0&lt;v1&lt;u1, un+1=un2+vn2un+vn, vn+1=un+vn2.

Q: Question 3. Let g, = nx1/n.2/a] and g = 0. Show that 9 lim Gn. Does the sequence (gn) converge…

A: Let gn=nℵ1n, 2n and g=0. Show that ∫02g ≠limn→∞∫02gn. The characteristic function ℵA is defined as…

Q: . Prove that if {fn} C C(K) converges pointwise and is equicontinuous on a compact set K, then it…

A:

Q: The current, I milliamperes, in a circuit is measured for various values of applied voltage V volts.…

A: The given problem is to find the best fit for the given recorded values and also find the suitable…

Q: 1. Let {9n} be a sequence of integrable functions which converges a.e. to an integrable function g.…

A: Solution: Given that gn is a sequence of integrable functions which converges a.e. to an integrable…

Q: (h) (3x)* (3.r)* 2k |1 k=1

A:

Q: Please help me problem 1(a), please as clear as possible for the proof

A: Given: