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Thread: Convergence on the real-line

  1. #1
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    Convergence on the real-line

    Hi, I need help with the following proof question for my stats course. It states:

    On the real line consider the collection of left-open/right-closed intervals:

    $\displaystyle I = \left\{ (a,b] : a,b \in \mathbb{R} \right\}$

    a) Write each of (0,1) and {0} explicitly as monotone limits of elements, $\displaystyle A_n$ in $\displaystyle I$.

    b) For the $\displaystyle A_n$'s chosen in a), determine the lower and upper sequences
    $\displaystyle B_n = \bigcap_{k \ge n }A_k , k = 1,2,...$ and
    $\displaystyle C_n = \bigcap_{k \ge n}A_k, n = 1,2,...$

    c) In general if $\displaystyle A_n$, n = 1,2,... is monotone prove that it does indeed converge and, explicitly, that
    $\displaystyle lim A_n = \bigcup A_n$ if $\displaystyle A_n \uparrow$ and
    $\displaystyle lim A_n = \bigcap A_n$ if $\displaystyle A_n \downarrow$

    d) Let $\displaystyle a_n = (-1)^n/n , n = 1,2,...$ so it is clear that $\displaystyle a_n \to 0 $
    For $\displaystyle A_n = (a_n,1], n = 1,2,...$ provide both its upper and lower sequences as in b) and thus prove whether or not $\displaystyle A_n$ converges.

    Thanks a lot. I'm trying really hard to wrap my head around this stuff. : )
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  2. #2
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    $\displaystyle \left( {0,1} \right] = \bigcap\limits_{n \in \mathbb{Z}^ + } {\left( {0,1 + \frac{1}{n}} \right)} \;\& \,\left\{ 0 \right\} = \bigcap\limits_{n \in \mathbb{Z}^ + } {\left[ {0,\frac{1}
    {n}} \right)} $
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  3. #3
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    I think I need a little bit more help answering these questions than just that. Thanks a lot though.
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  4. #4
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    Quote Originally Posted by AKTilted View Post
    I think I need a little bit more help answering these questions than just that. Thanks a lot though.
    In that case, you need a sit-down session with a live tutor.
    This is not the place to get that depth of help.
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  5. #5
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    "Get that depth of help"? I'm just looking for more help than just one line of math with zero explanation. If you could afford that I would be much more appreciative than receiving condescending responses like in your last post...
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