Introduction to Mathematical Thinking-Problem 9

最新推荐文章于 2020-02-02 15:00:21 发布

yuh_yeet

最新推荐文章于 2020-02-02 15:00:21 发布

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探讨了一组递减的区间序列AnA_nAn，其中每个区间都是前一个区间的子集，并且所有区间的交集为空集。通过设定An=(0,1n)A_n=(0,\frac{1}

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9.Give an example of a family of intervals $A_n$ such that $A_{n+1}\subset A_n$ for all $n$ and $\displaystyle \bigcap_{n=1}^\infty A_n=\emptyset$ .

Proof: Set $A_n=(0,\frac{1}{n})$ .
Then obviously, we have $A_{n+1}\subset A_n$ .
If $x$ is an element in $\displaystyle \bigcap_{n=1}^\infty A_n$ , then we can find a natural number $x$ such that $1 / n < x$ . So $x$ isn’t an element in $\displaystyle \bigcap_{n=1}^\infty A_n$ .
Therefore, we have found $A_n=(0,\frac{1}{n})$ such that $A_{n+1}\subset A_n$ for all $n$ and $\displaystyle \bigcap_{n=1}^\infty A_n=\emptyset$ .