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# Does positive real numbers contain a least element?

Does positive real numbers contain a least element? If so, then why?

Does every non empty subset of positive real numbers contain a least element? If so, why? If it is not, then why not?

I should prove both, by well ordering theorem, but I am not sure how to prove differently both question. Please help with these questions!

what about the set of all positive reals? Let $l$ be the least element. What can you say about $\frac{l}{2}$?

Here is a short proof that there is no least element.

Let $x\in(0,\infty)$ be the least positive real number. Since the interval is open, there exists some $\delta\in \Bbb{R}^+$ such that $x-\delta\ge 0$ and thus $\delta$ is smaller than $x$, yet a positive real number; $\Rightarrow\Leftarrow$.

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# Does positive real numbers contain a least element?

Does positive real numbers contain a least element? If so, then why?

Does every non empty subset of positive real numbers contain a least element? If so, why? If it is not, then why not?

I should prove both, by well ordering theorem, but I am not sure how to prove differently both question. Please help with these questions!

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what about the set of all positive reals? Let $l$ be the least element. What can you say about $\frac{l}{2}$?

for both questions? I used that way for the first one - chen yesterday

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