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Math Help - infimum of set proof

  1. #1
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    infimum of set proof

    i got such a set { \frac{5}{n} |n is a natural numbers}
    prove that inf { \frac{5}{n} |n is a natural numbers}=0
    n=1,2,3...
    i know that the limit is 0 when n goes large
    so the infimum is zero.
    but i need to prove it by definition

    i need to prove that 0 is the highest lower bound

    suppose 0 is not infimum and there is t which t>0
    and i need to desprove that t is the highest lower bound

    ?
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  2. #2
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    Re: infimum of set proof

    Quote Originally Posted by transgalactic View Post
    i got such a set { \frac{5}{n} |n is a natural numbers}
    prove that inf { \frac{5}{n} |n is a natural numbers}=0
    n=1,2,3...
    You know that 0 is a lower bound for that set.
    If c>0 show that there is some x is the set such that 0<x<c.
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  3. #3
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    Re: infimum of set proof

    i dont know how to show that there is a number in the set which is smaller then c

    ?
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  4. #4
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    Re: infimum of set proof

    Quote Originally Posted by transgalactic View Post
    i dont know how to show that there is a number in the set which is smaller then c
    If 0<c<1 there is a positive integer N
    such that N>\frac{5}{c} so \frac{5}{N}<c.
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  5. #5
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    Re: infimum of set proof

    thanks i got it
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