Let $A=\{1-1/n:n\in\mathbb{N}\}⊂\mathbb{Q}$
I want to proof the supremum and infimum of the set $A$ is. I can intuitively see that it is $sup(A)=1$ and $inf(A)=0$.
I know I need to proof that 1 is in the upper bound and 0 is in the lower bound, and then that it is the smallest upper and largest lower bound.
The upper bound of $A$ is $1$, as $1/n$ tends to 0 when $n$ gets larger. And the lower bound is $0$ for the same reason.
But how do I reason/proof that they are the smallest upper and largest lower bound?
Best Answer
You want to show that for all $x<1$, $x$ is not an upper bound of $A$, which means there exists some $a\in A$ such that $a>x$. Finding this $a$ will have to depend on $x$.
Similarly, you want to show that for all $x>0$, $x$ is not a lower bound of $A$, meaning there exists some $a\in A$ such that $a<x$. Finding this $a$ will be easy.