Is a continuous net of hyperreal numbers eventually constant? Let $(x_\alpha)$ be a net of hyperreal numbers indexed by the class of ordinals satisfying the condition that $x_\beta = {\lim}_{\alpha<\beta}(x_\alpha)$ for all limit ordinals $\beta$.  Then my question is, is $(x_\alpha)$ eventually constant?  If so, how quickly will it become constant?
And if the answer is no, does the answer become yes if we also add the condition that $(x_\alpha)$ is a convergent net?
 A: This is true not just in the hyperreal numbers but in any (set-sized) $T_1$ topological space.  Indeed, let $X$ be a $T_1$ space and let $\kappa>|X|$ be a regular cardinal.  Note then that any net $(y_\alpha)_{\alpha<\kappa}$ which converges to a point $y\in X$ must be eventually constant with value $y$.  Indeed, for each $z\in X\setminus\{y\}$ there is $\alpha_z<\kappa$ such that $y_{\alpha}\neq z$ for all $\alpha\geq\alpha_z$ (since $X$ is $T_1$).  Since $\kappa$ is regular and $\kappa>|X|$, we have $\sup_z \alpha_z<\kappa$ and then we must have $y_\alpha=y$ for all $\alpha\geq \sup_z \alpha_z$.
Now suppose $(x_\alpha)_{\alpha\in Ord}$ is a continuous net in $X$ which is not eventually constant.  We can then by transfinite recursion construct an increasing sequence of ordinals $(\alpha_\beta)_{\beta<\kappa}$ such that $x_{\alpha_{\beta+1}}\neq x_{\alpha_\beta}$ for each $\beta$.  Let $\gamma=\sup_\beta\alpha_\beta$. Then by continuity of the net, $(x_{\alpha_\beta})$ must converge to $x_\gamma$.  But this is impossible, since $(x_{\alpha_\beta})$ is not eventually constant.

As for "how quickly" it becomes constant, the argument shows that the net can only change values fewer than $\kappa$ times before becoming constant.  For the hyperreals, if your definition of them requires them to be countably saturated, you can actually take $\kappa=\omega$, since a sequence of length $\omega$ can never converge unless it is eventually constant due to countable saturation.  So, a continuous net indexed by the ordinals can only change values finitely many times before becoming constant.
