Let $A= $ {$10^n :n∈\mathbb Z$}. Find GLB$(A)$

Question

Can someone please explain to me how to do this? I can’t find any examples of this sort of question. Thanks.

Answer 1

We want to find $$\inf{\{10^n\,|\,n\in\mathbb{Z}\}}$$ Where $\inf{A}$ is the Infimum of a set $A$. As the function $f(x)=10^x$ is strictly increasing we can say that $$\inf{\{10^n\,|\,n\in\mathbb{Z}\}}=\lim_{x\to-\infty}10^x=0$$

Answer 2

$0$ is a lower bound of $A= ${$10^n:n\in\mathbb Z$}, because, for all $n \in \mathbb Z,$ $10^n>0$.

For any $\epsilon>0$, we can take $N<\log_{10}\epsilon,$ and then $10^N<\epsilon,$ so no $\epsilon>0$ is a lower bound of $A$.

Therefore, $0$ is the greatest lower bound of $A.$