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Sequence that converges to a point

  1. Dec 12, 2011 #1
    1.Problem Statement:

    If O is an open subset of ℝ does there exist a sequence in O that converges to x? Explain.

    2.Relevant equations

    3. The attempt at a solution

    So if I define a open subset of ℝ to be open if for all points x [itex]\in[/itex] O there exists a ε-neighborhood [itex]_{V}[/itex]ε (a) [itex]\subseteq[/itex]O.

    Then I would use pointwise convergence to prove that for each n[itex]\in[/itex] N, let fn be a function defined on a set A [itex]\subseteq[/itex] ℝ. The sequence fn of function converges pointwise on A to a function f : A → ℝ if for all x in A the sequence of real numbers fn(x) converges to f(x).
     
  2. jcsd
  3. Dec 12, 2011 #2
    You need to consider 3 cases:
    1. x is in O
    2. x is a boundary point of O
    3. x is neither in O nor a boundary point of O

    Should be straightforward from there
     
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