Let $C$ be a connected set in $\mathbb{R}$. Let $p$ be a limit point of $C$. Let $f:C\rightarrow \mathbb{R}$ be a function.
Suppose that;
$\exists \epsilon>0$ such that [ $\forall \delta>0, \exists x\in C$ such that $0
Now, define $A_n = \{x\in C|0
Then, it can be shown that $\forall m\in \omega, m\preceq A_n$.
Q1. Does this imply that $A_n$ is dedekind-infinite in ZF? If so, how?
Q2. $\{A_n\}$ defined above is a decreasing sequence and $\mathbb{R}$ is complete. I guess we can form a sequence $\{p_n\}$ in $C$ such that $p_n \rightarrow p \bigwedge p_n≠p \bigwedge \lim_{n\to\infty} f(p_n)≠q$. Can we? (in ZF)
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