I have little trouble with the following examples of sets to find its minimum, maximum, supremum and infimum. Sometimes it gets tricky if the example is not trivial. I know that here are many examples like that but I want an answer for those two. For this examples I will show my way of thinking, correct me please:
$ x = \big\{x:x=\frac{n-1}{n},n \in N^+\big\} $
$S=[0,1)\in R $
I'm using the following definitions that supremum is the smallest upper bound $x \in R$, and s is the element of a set S , $\forall \epsilon >0 , \exists s \in S:s>x-\epsilon , $ respectively the infimum is the greatest lower bound $\forall \epsilon >0 , \exists s \in S:s<x+\epsilon , $
- for $ n=1,2,3,4... , x=0,\frac{1}{2}, \frac{2}{3}, \frac{3}{4}... $ the values of x will never reach 1
So from my point of view, $min(S)=0$ and $ Inf (S)=0$ but $max(S)=\nexists $ and I think that $sup(S)$ does not exist, why I think that? From the definition, let us pick a number $s $ that would be greater than least upper bound minus a very small number but we don't know what is the smallest upper bound because the set is infinite
2.$min(S)=0$ and $max(S)=\nexists$ $Inf (S)=0$ but what with $Sup(S)=?$
If we pick a number $s=1 \notin S $ that for sure $1>x-\epsilon $ but the element s is not in the set so how to prove that for example $0.9999999>1-0.0000001$ , easily they gonna be equal but is it possible the left-hand side of the inequality to be grater or simply $Sup(S)=\nexists$