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Math Help - Proof With Definition of a Limit- Please help!

  1. #1
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    Proof With Definition of a Limit- Please help!

    Let l be an element of R, c be an element of R, c>0. Show that the following two statements are equivalent:
    a)for every epsilon>0 there exists N that's an element of the natural numbers for every n>or=N such that |xn-l|< epsilon
    b)for every epsilon>0 there exists N that's an element of the natural numbers for every n>or=N such that |xn-l|< (c)epsilon

    I came across this problem in my textbook while studying, but can't figure out how to approach it. Any help would be great!
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  2. #2
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    We may use the fact that if c,\epsilon>0, then both c\epsilon>0 and \frac{\epsilon}{c}>0.
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  3. #3
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    What if c is less than 1?
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  4. #4
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    That would be okay; the conclusions of

    \begin{aligned}<br />
c,\epsilon>0&\rightarrow c\epsilon>0\\<br />
c,\epsilon>0&\rightarrow \frac{\epsilon}{c}>0<br />
\end{aligned}

    would still follow.
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  5. #5
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    i've now tried the proof several times and still can't show that both statements imply each other, can you provide some more guidance?
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  6. #6
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    To prove (b) from (a), we may apply theorem (a) to the positive number \epsilon'=c\epsilon.
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