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: Closed set: definition

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    http://img527.imageshack.us/img527/6049/48193240ao5.png [Broken]

    I don't understand this proof. The first two lines are clear to me: the sequence xn is in F and F is closed so its complement is open so there is a ball with radius r around x in Fc.

    But I don't understand the last two lines. Of course there y larger then the radius od the ball but what's the relation with the converging sequence?
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Sep 21, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    If x_n -> x then the sequence x_n is eventually in O.
  4. Sep 21, 2008 #3
    But then you do not need the line that there are y which are greater than the radius and inside F, right?
  5. Sep 21, 2008 #4


    User Avatar
    Science Advisor

    Yes, you do. That's the whole point. Since {xn} converges to x, given any r> 0, there exist N such that if n> N, d(x, xn)< r. Taking r such that Br(x) is in O, it follows that xn, for n> N s in Br(x) so in O. IF, as in the hypothesis, all xn are in F, then some members of O (they "y" they mention) are in F, a contradiction.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook