1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Constructing an interval by uniting smaller intervals

  1. May 10, 2013 #1
    1. The problem statement, all variables and given/known data
    We have [itex]C_n = [1-\frac{1}{n},2-\frac{1}{2n}] [/itex] and [itex]C = C_1 \cup C_2 \cup C_3 \cup ...[/itex] and are asked to describe the interval C and then prove that it is actually what we say it is.

    2. Relevant equations

    3. The attempt at a solution

    I am guessing that [itex] C = [0,2) [/itex] and to prove this, I need to show that the infimum and minimum is 0, supremum is 2, maximum does not exist.

    Am I right in describing this interval? And what other features of this interval would I need to prove in order to prove my description is valid?
  2. jcsd
  3. May 10, 2013 #2


    User Avatar
    Science Advisor

    Why do you think 2 is not included in the interval? What is the limit of 2-(1/2n)? If 2 is not included,

    in C, can 2- 1/2n converge to 2?
  4. May 10, 2013 #3
    I don't really understand what you mean. 2 cannot be in C because 2-1/2n -> 2 as n -> ∞, so C_n = (1,2) as n -> ∞, wouldn't it?
  5. May 10, 2013 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I doubt that Bacle2 is saying that 2 is in interval C. He's just asking how to prove that you can have both
    2 is not in C.


    ##\displaystyle \lim_{n\to\infty}(2-1/(2n))=2 ## ?​
  6. May 10, 2013 #5
    Well, now I have no idea.

    How else could I show that 2 is not in C? Unless 2 IS in C...
  7. May 10, 2013 #6
    If I make [itex]A[/itex] the set of all maxima of all [itex] C_n = \{ 2-\frac{1}{2n}\}[/itex] for all natural numbers n,

    Then the maximum of this set [itex]A[/itex] does not exist since it is infinite. However the least upper bound would be the limit = 2. Am I correct in thinking this?

    Then sup(C) = sup(A) = 2. Is this reasoning correct?
  8. May 10, 2013 #7


    User Avatar
    Science Advisor

    Sorry if my statement was not clear. What I meant was that if 2 were included in the union, then it would be in some of the intervals, by def. of union. Then, re this last, the issue of convergence ,and (2-1/2n) being strictly increasing, come into play.
    Last edited: May 10, 2013
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted