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Math Help - Finding inf

  1. #1
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    Finding inf

    Let A=\left\{\dfrac1n:n\in\mathbb N\right\}, B=(-1,0] and d(x,y)=|x-y|, then find d(A,B)=\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,d(a,b)=\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,\left| \dfrac{1}{n}-b \right|.

    I know it's zero, but don't know exactly how to prove it.

    Thanks for the help!
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  2. #2
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    Quote Originally Posted by Connected View Post
    Let A=\left\{\dfrac1n:n\in\mathbb N\right\}, B=(-1,0] and d(x,y)=|x-y|, then find d(A,B)=\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,d(a,b)=\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,\left| \dfrac{1}{n}-b \right|.

    I know it's zero, but don't know exactly how to prove it.

    Thanks for the help!

    Since \displaystyle{0\in B\mbox{ and } \frac{1}{n}\xrightarrow [n\to\infty]{}0 , you have that

    \displaystyle{\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,d(a,b)=\underset{ b\in B}{\mathhop{\inf}} \lim\limits_{n\to\infty}\left|\frac{1}{n}-b\right| .

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Since \displaystyle{0\in B\mbox{ and } \frac{1}{n}\xrightarrow [n\to\infty]{}0 , you have that

    \displaystyle{\underset{\begin{smallmatrix} <br />
 a\in A \\ <br />
 b\in B <br />
\end{smallmatrix}}{\mathop{\inf }}\,d(a,b)=\underset{ b\in B}{\mathhop{\inf}} \lim\limits_{n\to\infty}\left|\frac{1}{n}-b\right| .

    Tonio
    Okay, this seems to be a property or something, which is exactly?
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  4. #4
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    Quote Originally Posted by Connected View Post
    Okay, this seems to be a property or something, which is exactly?

    No...it just follows from the definition. You could as well choose an arbitrary \epsilon > 0 and prove that

    there exist a\in A\,,\,\,b\in B\,\,s.t.\,\,d(a,b)\geq\epsilon , and again by definition 0 is the infimum.

    Tonio
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