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Convergence of sequences

  1. May 5, 2007 #1
    1. The problem statement, all variables and given/known data
    Does
    A subsequence of a sequence X converges to a point in I => The sequence X in I converges to a point in I
    ?


    3. The attempt at a solution
    I think yes because the subsequence is the sequence itself minus a few finite number of points. Since they both are in the same set I, I can't see why not.
     
    Last edited: May 5, 2007
  2. jcsd
  3. May 5, 2007 #2

    Office_Shredder

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    A subsequence can be lacking an infinite number of points in I. If a sequence is 1,-1,1,-1,1,-1,...

    the subsequence

    1,1,1,1,1...

    certainly converges. You can tell me what you think about the statement
     
  4. May 5, 2007 #3
    Good example.

    The question should be
    Does
    A subsequence of a sequence X converges to a point in I <= The sequence X in I converges to a point in I
    ?

    Now it should be yes.

    But we genearlly refer to seq and subseq as containing an infinite number of points.
     
  5. May 5, 2007 #4

    quasar987

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    If [tex]f:\mathbb{N}\rightarrow I[/tex] is a sequence in a set I, then a subsequence of f is a sequence of the form h = f o g, where [tex]g:\mathbb{N}\rightarrow\mathbb{N}[/tex] is a strictly increasing sequence of natural numbers.

    I like to think of g as a discriminating function that picks which guys from f it wants in its kickball team.. or which girls does the Maharajah wants in its harem, or... any such pictorial analogy to remember the definition.
     
    Last edited: May 5, 2007
  6. May 5, 2007 #5

    quasar987

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    And it goes farther too: The sequence X in I converges to a point y in I <=> Every subsequence of X converges to y.
     
  7. May 6, 2007 #6
    It a little subtle. Every is essential. I didn't have every in my original statement so no if and only if condition.
     
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