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Let $S=[0,1]$. If $x$ and $y$ are in $S$ with $x\neq y$. How can we show that there are $m,n\in \mathbb{N}$ such that $x< \dfrac{m}{2^n} <y$. Can the Archimedean Property be used to prove this? If yes, could anyone provide me an insight to do this?

Let's call a set "Pseudo compact" if it has the property that every closed cover (a cover consisting of closed sets) have a finite subcover. I have a little ides on "Anti-Compact: subsets. Does Pseudo-Compact in this case the same as Anti-Compact ? Then how can we describe the "P...

I am aware a set is Bounded if it has both upper and Lower bounds and i know what a Limit point of a set is but i am finding it hard to show that If $S\subset\mathbb{R}$ be a "bounded infinite set", then $S'\neq\varnothing$ .