Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of a sequence of sets: hypercubes.

  1. Dec 20, 2005 #1


    User Avatar
    Homework Helper

    Ok, so I would like to formally settle this one. I have a sequence of sets [itex]C_{N}^{n}\subset\mathbb{R} ^{n}, N\in\mathbb{Z} ^{+}[/itex] defined as follows

    [tex]C_{N}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\} [/tex]

    I would like to prove that [itex]C_{N}^{n}\rightarrow\mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1) \mbox{ as }N\rightarrow \infty[/itex] (through integer values.)

    I have done some study of limits of sequences of sets: most fruitful has been Measure Theory, by Halmos, in which the limit of a sequence of sets is defined as the set [itex]A = \liminf A_{n} = \limsup A_{n}[/itex], where the upper and lower limits of a sequence of sets [itex]\left\{ A_{n} \right\} [/itex] are defined by

    [tex]\liminf A_{n} = \bigcup_{j=1}^{\infty} \bigcap_{k=j}^{\infty} A_{k}[/tex]


    [tex]\limsup A_{n} = \bigcap_{j=1}^{\infty} \bigcup_{k=j}^{\infty} A_{k}[/tex]

    But, I hadn't heard of that when I started playing with the above sets. It is easy to see that, as "limit equations," the following are tennable:

    [tex]Q^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\}= \mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1) [/tex]

    and, in fact, the sets

    [tex]Q_{d}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n-d \right\}[/tex]

    possess that property of describing precisely the d-dimensional content (less lower dimensional boundaries) of the hypercube with verticies at [itex](\pm 1, ..., \pm 1) [/tex]; that is to say that the sets [itex]Q_{0}^{n},Q_{1}^{n},...,Q_{n-2}^{n},Q_{n-1}^{n},\mbox{ and }Q_{n}^{n}[/itex] describe the vertices, edges..., ridges, facets, and hypervolume of said n-dimensional hypercube, respectively. And since [itex]Q_{i}^{n}\cap Q_{k}^{n}=\emptyset, \mbox{ for }j\neq k[/itex], and [itex] \bigcup_{k=1}^{n} Q_{k}^{n} = Q^{n}[/itex] the above sets provide a strataification of said hypercube.

    So my quesion is: "Is the later "limit equation" interpetation given consistent with the notion of limits of sequences of sets given by Halmos?"
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

Similar Discussions: Limit of a sequence of sets: hypercubes.
  1. Limit of a sequence (Replies: 28)

  2. Limit of sequence (Replies: 1)

  3. Limits of a sequence (Replies: 1)