Let $(x_\alpha)$ be a net of real numbers indexed by the class of ordinals, converging to some real number $x$. Then my question is, for any ordinal $\beta$, does the sequence $(x_\gamma)$ indexed by the set of ordinals less than $\beta$ also converge to $x$? Or can it diverge or converge to a different value?
If the answer is no, does anyone know of a counterexample?
No. For all n in N, let x$_{2n}$ = 0, x$_{2n+1}$ = 1,
and for all $\alpha$ >= $\omega_0$, let x$_{\alpha}$ = 0.
Continue this ordinal sequence as far as one wishes.