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Math Help - limit points

  1. #1
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    limit points

    Hello math lover :

    I face some difficulties in solving the problem attached in a file. Can
    you help me please.

    Thank you in advance
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  2. #2
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    Given any \epsilon> 0 If 0< x< \epsilon, then \frac{1}{\epsilon}< \frac{1}{x}< \infty which includes an infinite number of periods of sine. That is, sin(1/x) will take on any value between -1 and 1 and is within \epsilon of any point in that set.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by fuzzy topology View Post
    Hello math lover :

    I face some difficulties in solving the problem attached in a file. Can
    you help me please.

    Thank you in advance
    d\left((0,1),\left(x,\sin\left(\tfrac{1}{x}\right)  \right)\right)=\sqrt{x^2+\left(1-\sin\left(\frac{1}{x}\right)\right)^2}. By the Archimedean principle there exists some n\in\mathbb{N} such that \frac{1}{n}<\varepsilon and so \frac{1}{2\pi n+\frac{\pi}{2}}<\varepsilon. And so d\left((0,1),\left(\tfrac{1}{2\pi n},\sin\left(2\pi n\right)\right)\right)=\sqrt{\left(\tfrac{1}{2\pi n}\right)^2+\left(\sin\left(2\pi n+\tfrac{\pi}{2}\right)-1\right)^2}=\frac{1}{2\pi n}<\varepsilon
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  4. #4
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    Thaaaaaaaaank you very much dexel28.
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