QuestionQuestion

Transcribed TextTranscribed Text

Question 5 Throughout this question we use the Euclidean metric for both [0, 1] and R. (a) For n ∈ N, define a function fn : [0, 1] → R by fn(x) = Xn i=0 2 −i (1 + sin(2ix)). (i) Prove that sequence (fn) has a pointwise limit, f : [0, 1] → R. Hint: use the monotone convergence theorem. (ii) By showing that for each n ∈ N, the function fn is Lipschitz, prove that each function fn is continuous. (iii) Determine whether the pointwise limit of the sequence (fn) is continuous. (b) For N ∈ N, dene FN : C[0, 1] → R by FN (f) = 1 N PN i=1 f( i N ). Show that FN is (dmax, d)-continuous on C[0, 1], where d is the Euclidean metric for R. Question 6 Dene a distance function on the plane, d : R 2 × R 2 → R by d(x, y) = ( d (2)(x, 0) + d (2)(y, 0), if x 6= y 0, if x = y. (a) Prove that d is a metric on R 2 . (b) (i) Let (xn) be a sequence of points R 2 . Show that (xn) is a d-convergent sequence if, and only if, either (xn) is eventually constant or (d (2)(xn, 0))n∈N is a real null sequence (in which case the d-limit is 0). (Recall that d (2) denotes the Euclidean metric for the plane.) (ii) Find Cl(R2,d)(A) when A = {(x1, x2) ∈ R 2 : x1 > 0}.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    50% discount

    Hours
    Minutes
    Seconds
    $28.00 $14.00
    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Advanced Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats