Limit of a sequence of sets: hypercubes.

by benorin
Tags: hypercubes, limit, sequence, sets
 HW Helper P: 1,007 Ok, so I would like to formally settle this one. I have a sequence of sets $C_{N}^{n}\subset\mathbb{R} ^{n}, N\in\mathbb{Z} ^{+}$ defined as follows $$C_{N}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\}$$ I would like to prove that $C_{N}^{n}\rightarrow\mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1) \mbox{ as }N\rightarrow \infty$ (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 $A = \liminf A_{n} = \limsup A_{n}$, where the upper and lower limits of a sequence of sets $\left\{ A_{n} \right\}$ are defined by $$\liminf A_{n} = \bigcup_{j=1}^{\infty} \bigcap_{k=j}^{\infty} A_{k}$$ and $$\limsup A_{n} = \bigcap_{j=1}^{\infty} \bigcup_{k=j}^{\infty} A_{k}$$ 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: $$Q^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\}= \mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1)$$ and, in fact, the sets $$Q_{d}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n-d \right\}$$ possess that property of describing precisely the d-dimensional content (less lower dimensional boundaries) of the hypercube with verticies at $(\pm 1, ..., \pm 1) [/tex]; that is to say that the sets [itex]Q_{0}^{n},Q_{1}^{n},...,Q_{n-2}^{n},Q_{n-1}^{n},\mbox{ and }Q_{n}^{n}$ describe the vertices, edges..., ridges, facets, and hypervolume of said n-dimensional hypercube, respectively. And since $Q_{i}^{n}\cap Q_{k}^{n}=\emptyset, \mbox{ for }j\neq k$, and $\bigcup_{k=1}^{n} Q_{k}^{n} = Q^{n}$ 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?"

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