4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...

Question

Question

4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn → Irl uniformly on I-1, 1]. Note that the limit function I

math Advanced-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

Ang - even data. es 1 xl on (1,

Add a comment

Answer 2

Similar Homework Help Questions

1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx)...

1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x...

For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...

4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...

Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose...

Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|&l...

(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|< M, for al E [0, 1], nEN. Prove that there is a subsequence {fng} converging uniformly to a function f e C((0, 1]) on 0, 1]. (1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|
(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a,...

(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a, b) such (a, b), then f is also uniformly continuous on (a, b) that f funiformly on en dr 0. (11) Show that lim n-+o0 e (10) A G.. 11d

Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that...

Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges...

Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
. Prove that sequence in Example 6.2.2 (i) on p.174 converges uniformly to r on any inteval [a, b]. Prove that the...

. Prove that sequence in Example 6.2.2 (i) on p.174 converges uniformly to r on any inteval [a, b]. Prove that the convergence cannot be uniform on [0, 0o) J() d tel argue thau Jn J Exercise 6.2.6. Assume fn → f on a set A. Theorem 6.2.6 is an example of a typical type of question which asks whether a trait possessed by each fn is inherited by the limit function. Provide an example to show that all of...

3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2...

3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2 Ils le = (150)Par)". Let (fn)nen be a sequence of functions in R, defined by 1<3 fn(x) = 1 VI V 0 < (a) Prove that (fn)nen is Cauchy. (b) Prove that (fn) does not converge in R([0, 1]). (Note: If it did, then what must the limit function be? Can this candidate function be in R?)

4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...

Homework Answers

Add Answer to:
4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...

Post as a guest

Earn Coins

1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx)...

For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x...

4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...

Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose...

(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|&l...

(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a,...

Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that...

Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges...

. Prove that sequence in Example 6.2.2 (i) on p.174 converges uniformly to r on any inteval [a, b]. Prove that the...

3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2...

4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...

Homework Answers

Add Answer to: 4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...

Post as a guest

Earn Coins

Add Answer to:
4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn...