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Closed set: definition

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    [​IMG]

    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?
     
  2. jcsd
  3. Sep 21, 2008 #2

    morphism

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    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

    HallsofIvy

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    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.
     
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