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Math Help - Convergence

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    MHF Contributor harish21's Avatar
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    Convergence

    The sequence { 1/n } converges to 0 and that it does not converge to any other number. Using this fact prove that NONE of the following assertions is equivalent to the definition of convergence of a sequence {an} to the number a.

    a. for some e>0, there is an index N such that |an -a| < e for all indices n>=N

    b. for each e>0 and each index N, |an - a| < e for all indices n>=n

    Can anyone give me a hint or an outline about this. i am totally clueless about it.
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    Quote Originally Posted by harish21 View Post
    The sequence { 1/n } converges to 0 and that it does not converge to any other number. Using this fact prove that NONE of the following assertions is equivalent to the definition of convergence of a sequence {an} to the number a.

    a. for some e>0, there is an index N such that |an -a| < e for all indices n>=N
    Show that this is true for a_n= \frac{1}{n}, a= 1, and e= 2. (The point is that this says "for some e> 0" rather than "for all e> 0".)

    b. for each e>0 and each index N, |an - a| < e for all indices n>=n
    Did you mean "n> N"? Show that this is NOT true if you take a_n= \frac{1}{n}, a= 0, e= .001, and N= 1. (The point is that this says "for each index N" rather than "for some index N".)

    Can anyone give me a hint or an outline about this. i am totally clueless about it.
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    MHF Contributor harish21's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    Show that this is true for a_n= \frac{1}{n}, a= 1, and e= 2. (The point is that this says "for some e> 0" rather than "for all e> 0".)


    Did you mean "n> N"? Show that this is NOT true if you take a_n= \frac{1}{n}, a= 0, e= .001, and N= 1. (The point is that this says "for each index N" rather than "for some index N".)

    Yes that was supposed to be n>=N in b. Thank you.

    Likewise there is one more question that states there is an index N such that for every number e>o |an - a| < e for all indices n>=N. What should I show here?
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