E = {x/2^y:x,y elemnts of N}
prove inf(E) = 0.
(please use contrapositive)

Question

E = {x/2^y:x,y elemnts of N}
prove inf(E) = 0.
(please use contrapositive)

kinggeorge · Answer

So I'm going to assume that $0\in\mathbb{N}$, otherwise I don't think this could be true.

kinggeorge · Answer

I was wrong, we don't have to assume 0 is in the natural numbers. There is a more elegant way. So we need to start by assuming $\inf(E)\neq0$. We have two cases Case 1: Suppose $\inf(E)<0$. That means that there exists some $a\in E$ such that $a<0$. However, $x$ and $2^y$ are always non-negative since $\mathbb{N}$ is non-negative. This is a contradiction, so $\inf(E)\geq0$. Case 2: Let $a=\inf(E)>0$. Then $a>0$, and given any $b\in E$, $a\le b$. Now, let $$\Large a=\frac{r}{2^s}$$Notice that $$\Large \frac{r}{2^{s+1}}< \frac{r}{2^s}$$Hence, $a$ is not $\inf(E)$, so $\inf(E)\leq 0$. It follows that $\inf(E)=0$.

kinggeorge · Answer

Let me know if you need help understanding certain steps.

zzr0ck3r · Answer

word, good idea. TY