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Math Help - Every infinite bounded set of R^n has an accumulation point

  1. #1
    Senior Member Pinkk's Avatar
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    Every infinite bounded set of R^n has an accumulation point

    So I have a general idea of what the proof should like (construct a sequence in S, since it is bounded the sequence has a convergent subsequence, and so the limit of that subsequence is an accumulation point). However, I am having a hard time construction such a sequence such that none of the terms of the subsequence are the limit point. How would I go about doing this. I considered using the supremum of S as the limit point but if the supremum is in S, there is no guarantee that every single term of a constructed sequence that converges to the supremum has terms in which none of the terms are equal to the supremum itself. Any help would be appreciated, thanks.
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  2. #2
    Senior Member Pinkk's Avatar
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    Okay, so proving this in \mathbb{R} is pretty easy but carrying that over to \mathbb{R}^{n} is another thing altogether. I am tempted to just apply the infinite and bounded condition to every component of each term in a sequence in \mathbb{R}^{n}, but just because the set is infinite in \mathbb{R}^{n} doesn't mean that if we look at each component and the set of all possible elements for that component is infinite, for instance the set (x,y,z) where x = y = 1 and z in [-1,1] is bounded and infinite but there are not infinitely many choices for each component. How do I resolve this?
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  3. #3
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    Since S is bounded, then there is a rectangle R_1=[a_{11},b_{11}]\times[a_{12},b_{12}]\times\cdots\times[a_{1n},b_{1n}] which includes S. Partition this rectangle into 2^n equally-sized rectangles. In particular,

    P_i=\{I_1\times I_2\times\cdots\times I_n:I_j\in\{[a_{ij},(a_{ij}-b_{ij})/2],[(a_{ij}-b_{ij})/2,b_{ij}]\}\},

    where i=1.

    Since there are finitely many ( 2^n) rectangles in P_i, at least one must contain infinitely many points of S. Choose that one, and denote it R_2=[a_{21},b_{21}]\times[a_{22},b_{22}]\times\cdots\times[a_{2n},b_{2n}].

    Repeat as needed, to build a sequence \{R_i\}_{i=1}^\infty Show that it converges to an accumulation point.
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