Let $A\subset\mathbb R$. Show for each of the following statements that it is either true or false.
- If $\min A$ and $\max A$ exist then $A$ is finite.
- If $\max A$ exists then $A$ is infinite.
- If $A$ is finite then $\min A$ and $\max A$ do exist.
- If $A$ is infinite then $\min A$ does not exist.
My attempts so far:
This statement is wrong. Let $A=[a;b]\cap\mathbb Q\subset\mathbb R$ with $a
This statement is wrong. Let $A=(0;n]\cap\mathbb N\subset\mathbb R$ with $n\in\mathbb N$ and therefore $A$ is bounded by $n$ with $\max A=n$. Assume $A$ would be infinite; with $|A|=n-1$ follows, that $A$ must be finite contrary to the assumption that it is infinite.
???
This statement is wrong. Let $A=\mathbb N\subset\mathbb R$ and we know that $A$ is infinite. However $A$ has a lower bound and even a minimal element where $\min A=1$ in constrast to the assumption that it does not exist.
I would like to know whether my attempts for (1), (2) and (4) are plausible or incomplete. Furthermore I need some hints on how to prove (3).