# infimum of set proof

• Aug 30th 2011, 05:49 AM
transgalactic
infimum of set proof
i got such a set {$\displaystyle \frac{5}{n}$ |n is a natural numbers}
prove that inf {$\displaystyle \frac{5}{n}$ |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

?
• Aug 30th 2011, 06:23 AM
Plato
Re: infimum of set proof
Quote:

Originally Posted by transgalactic
i got such a set {$\displaystyle \frac{5}{n}$ |n is a natural numbers}
prove that inf {$\displaystyle \frac{5}{n}$ |n is a natural numbers}=0
n=1,2,3...

You know that 0 is a lower bound for that set.
If $\displaystyle c>0$ show that there is some $\displaystyle x$ is the set such that $\displaystyle 0<x<c$.
• Aug 30th 2011, 06:34 AM
transgalactic
Re: infimum of set proof
i dont know how to show that there is a number in the set which is smaller then c

?
• Aug 30th 2011, 07:16 AM
Plato
Re: infimum of set proof
Quote:

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

If $\displaystyle 0<c<1$ there is a positive integer $\displaystyle N$
such that $\displaystyle N>\frac{5}{c}$ so $\displaystyle \frac{5}{N}<c$.
• Aug 30th 2011, 08:59 AM
transgalactic
Re: infimum of set proof
thanks i got it