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$\forall i\in\mathbb{N}^+,A_i\subset \mathbb{R}$,and $A_i=[i,\infty)$.Then what does

$\bigcap_{i=1}^{\infty}A_i$

mean?

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    it is $\emptyset$. Since for any $x\in R^+$, there is a integer $k\ge x$.2012-05-05

2 Answers 2

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It means what intersection always means: the set of things that belong to every $A_i$, $i=1,2,3,\dots$ . Suppose that $x\in\Bbb R$ and $n\in\Bbb N^+$. Then $x\in A_n$ if and only if $x\ge n$. Thus, $x\in\bigcap_{n=1}^\infty A_n$ if and only if $x\in A_n$ for all $n\in\Bbb N^+$, if and only if $x\ge n$ for all $n\in\Bbb N^+$. But there is no real number $x$ such that $x\ge n$ for all $n\in\Bbb N^+$, so $\bigcap_{n=1}^\infty A_n=\varnothing$.

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It means the set of all points that is in each $[i,\infty)$ for all $i$. There are so such points since if $x\in\mathbb{R}$ then $x\notin [\lceil x\rceil+1,\infty)=A_{\lceil x\rceil+1}$

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    Meet you again.Thanks for your help2012-05-05