Let $(x_\alpha)$ be a net of real numbers indexed by the class of ordinals, satisfying the following conditions:
- $x_{\alpha+1}=f(x_\alpha)$ for all ordinals $\alpha$, where $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$
- $x_\beta = {\lim}_{\alpha<\beta}(x_\alpha)$ for all limit ordinals $\beta$
- $(x_\alpha)$ converges to $x$ for some real number $x$.
Then my question is, is it necessarily true that $f(x)=x$? It's clearly true if $f$ is continuous, but what about the general case?
And if it is true, is it also true for nets indexed by ordinals less than some sufficiently large ordinal, like $\omega_1$ or $2^{\aleph_0}$ (assuming the axiom of choice)? This is a follow-up to my question here, by the way.
Best Answer
Yes, for a kind of dumb reason. At limit ordinals of uncountable cofinality, condition (2) will never hold except in trivial cases. Indeed, if $\beta$ is a limit ordinal, then for each $n\in\mathbb{N}$ there is $\alpha_n<\beta$ such that $|x_{\alpha}-x_\beta|<1/n$ for all $\alpha\geq\alpha_n$. If $\beta$ has uncountable cofinality, then $\sup_n\alpha_n<\beta$, and we must have $x_\alpha=x_\beta$ for all $\alpha$ such that $\sup_n\alpha_n\leq \alpha<\beta$. That is, $(x_\alpha)_{\alpha<\beta}$ is eventually constant, and in particular this means $f(x_\beta)=x_\beta$.
In particular, taking $\beta=\omega_1$, this shows that you must reach a fixed point at some countable ordinal.
There's nothing particularly important about $\mathbb{R}$ here--the same argument would work in any first countable $T_1$ space. Or, it would work in any $T_1$ space if you replace "uncountable cofinality" with "cofinality greater than the minimum size of a neighborhood base at each point".