Register to reply

Limit of a sequence of sets: hypercubes.

by benorin
Tags: hypercubes, limit, sequence, sets
Share this thread:
Dec20-05, 03:30 PM
HW Helper
P: 1,025
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?"
Phys.Org News Partner Science news on
Scientists discover RNA modifications in some unexpected places
Scientists discover tropical tree microbiome in Panama
'Squid skin' metamaterials project yields vivid color display

Register to reply

Related Discussions
Limit of a sequence Calculus & Beyond Homework 2
Limit of a sequence Calculus & Beyond Homework 10
Limit of sequence equal to limit of function Calculus 1
Limit of a sequence... Precalculus Mathematics Homework 11
Limit of a sequence Introductory Physics Homework 2