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Math Help - subsequence definition question..

  1. #1
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    subsequence definition question..

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  2. #2
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    Let X_n = x_{r_n}.
    Now, \liminf x_n \leq \limsup x_n \leq \limsup X_n = \lim X_n

    The reason why \limsup X_n = \lim X_n is because \{X_n\} is convergent.

    The rest of them are similar.
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  3. #3
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    <br />
\limsup X_n = \lim X_n<br />

    this expression is not correct
    X_n is bounded
    the limit of the sequence doesnt equal the limsup

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  4. #4
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    Quote Originally Posted by transgalactic View Post
    <br />
\limsup X_n = \lim X_n<br />

    this expression is not correct
    X_n is bounded
    the limit of the sequence doesnt equal the limsup

    The sequence X_n is convergent.
    Therefore, its limit matches its limit superior.
    Look at the theorems you learned.
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  5. #5
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    Xn is not convergent
    its bounded
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  6. #6
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    Quote Originally Posted by transgalactic View Post
    Xn is not convergent
    its bounded
    Answer:
    Quote Originally Posted by ThePerfectHacker View Post
    Let X_n = x_{r_n}.
    According to your work X_n is convergent. TPH's X_n is not the same as your X_n. He is talking about the subsequence which is convergent...
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  7. #7
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    <br />
\liminf x_n \leq \limsup x_n \leq \limsup X_n = \lim X_n<br />

    why
    <br />
limsupx_n \leq \limsup X_n<br />
    X_n is a subsequence which converges
    the definition of limsup
    is being the sup of the limit group of every subsequence

    you cant do limsup X_n
    by doing that you split this subsequence into a smaller convergent subsequences
    and take the Sup of their limits

    which has nothing to do with limsup x_n
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  8. #8
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    Quote Originally Posted by transgalactic View Post
    why
    <br />
limsupx_n \leq \limsup X_n<br />
    If A,B are non-empty bounded subsets with A\subseteq B then \sup (B) \leq \sup (A).

    Now, \{ x_{r_k} | k\geq n\} \subseteq \{ x_k | k\geq n\}.
    Therefore, \sup \{ x_k | k\geq n\} \leq \sup \{ x_{r_k} | k\geq n\}.
    Thus, \lim \sup \{x_k |k\geq n\} \leq \lim \sup \{ x_{r_k}| k\geq n\}.
    This gives, \limsup x_n \leq \limsup X_n
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