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Math Help - Am I allowed to do this?

  1. #1
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    Am I allowed to do this?

    I'm trying to construct an infinite sequence that converges to zero but has a positive (nonzero) limit supremum. Here's what I'm attempting to do:

    Let a_j = 1 if j = n+1 and 0 otherwise. Then the limit supremum equals (lim n -> inf) sup {a_n, a_n+1, a_n+2, ...}
    = (lim n -> inf) sup {0, 1, 0, 0, ...}
    = 1

    Am I allowed to do that, or is that "cheating"? If so, what's a workaround that would suffice?
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  2. #2
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    Re: Am I allowed to do this?

    Firstly, I don't think such a sequence can possibly exist.

    Secondly, I'm not sure what your sequence is... Is n fixed? Is j fixed?
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  3. #3
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    Re: Am I allowed to do this?

    Quote Originally Posted by Gusbob View Post
    Firstly, I don't think such a sequence can possibly exist.
    Ah OK. I didn't think so, but I really don't know how to verify this.

    Secondly, I'm not sure what your sequence is... Is n fixed? Is j fixed?
    IDK. I was just thinking about how to make what appears to be an impossible condition possible.
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  4. #4
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    Re: Am I allowed to do this?

    Quote Originally Posted by phys251 View Post
    Ah OK. I didn't think so, but I really don't know how to verify this.
    Check:

    A sequence a_n converges if and only if

    \liminf a_n = \lim a_n =\limsup a_n

    This should be obvious by definition.
    Thanks from phys251
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  5. #5
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    Re: Am I allowed to do this?

    Quote Originally Posted by Gusbob View Post
    Check:

    A sequence a_n converges if and only if

    \liminf a_n = \lim a_n =\limsup a_n

    This should be obvious by definition.
    Ah. That's what I thought. Thanks.
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