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Thread: Corollary on Irrational Number

  1. #1
    Member aldrincabrera's Avatar
    Jun 2011
    Dumaguete City

    Cool Corollary on Irrational Number

    ,.hello everyone,.,i need ur help agen with this,.,

    Let epsilon>0 be an irrational number and let z>0. then there exists a natural number m such that the irrational number epsilon/m satisfies 0<(epsilon/m)<z.

    ,.i somehow understand the proof on the book but i was having a hard time proving that

    epsilon/m is an irrational number,.,can anyone give me a hand with this???any hint will do,.,.thnx a lot[IMG]file:///C:/Users/User/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif[/IMG]
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  2. #2
    MHF Contributor

    Apr 2005

    Re: Corollary on Irrational Number

    Your image did not load. However, given any irrational number, $\displaystyle \epsilon> 0$, there exist a real number, r, greater than $\displaystyle \epsilon$, $\displaystyle \epsilon+ 1$, for example. Given that, let z= r/m where m is any natural number. Then $\displaystyle 0< \epsilon< r= mz$ and so $\displaystyle 0< \epsilon/m< z$.
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