# Thread: infimum of set proof

1. ## 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

?

2. ## Re: infimum of set proof

Originally Posted by transgalactic
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.

3. ## Re: infimum of set proof

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

?

4. ## Re: infimum of set proof

Originally Posted by transgalactic
i dont know how to show that there is a number in the set which is smaller then c
If $0 there is a positive integer $N$
such that $N>\frac{5}{c}$ so $\frac{5}{N}.

5. ## Re: infimum of set proof

thanks i got it