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Math Help - countably many intersection of sets

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    countably many intersection of sets

    How would you rigorously show
     \cap_{n=1}^{\infty}[a-\frac{1}{n},\infty)= (a,\infty) for each a\in\mathbb{R}
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by willy0625 View Post
    How would you rigorously show
     \cap_{n=1}^{\infty}[a-\frac{1}{n},\infty)= (a,\infty) for each a\in\mathbb{R}
    Clearly [a,\infty)\subseteq \left[a-\tfrac{1}{n},\infty\right),\text{ }\forall n\in\mathbb{N}. Now, suppose that b<a\in\bigcap_{n=1}^{\infty}\left[a-\tfrac{1}{n},\infty\right). We clearly have that b<a\implies a-b>0 and thus by the Archimedean principle there exists some m\in\mathbb{N} such that \frac{1}{m}<a-b\implies b<a-\frac{1}{m}\implies b\notin\left[a-\tfrac{1}{m},\infty\right). It follows that b\notin\bigcap_{n=1}^{\infty}\left[a-\tfrac{1}{n},\infty\right)
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