Texts on limits of ordinal sequences

Question

Let $\alpha$ be a limit ordinal, and let $f$ be a function from $\alpha$ to $\mathbb{R}$. Has anyone or any book or text defined the notion of limits for general limit ordinals, not just the limit ordinal $\omega$? For example, let $\alpha=\omega^2$. Consider two sequences on $\alpha$, the first being, $1,1/2,1/4,1/8,...1,1/2,1/4,1/8,...$. Basically, it is just $\omega$ copies of the sequence $1,1/2,1/4,1/8,...$ The second sequence is $1,1/2,1/4,1/8,...,1/2,1/4,1/8,1/16,...,1/4,1/8,1/16,1/32,...$ Basically, it is just the sequence $1,1/2,1/4,1/8,...$ multiplied by $1/2$ over and over again. It is easy to see, and can be proven almost as easily, that the first sequence does not have a limit, while the second sequence has limit $0$. So, my question is, has any book or text defined the notion of a limit of a sequence of a general limit ordinal? I used the example of $\mathbb{R}$, but it could also be a more complicated space, like $\mathbb{C}$, or the space of continuous functions on $\mathbb{R}$.

Answer 1

You can just apply the normal definition of convergence for sequences, but replace naturals with ordinals. Under this definition, the sequence on $\alpha$ will converge to $0$. Note that the reals are separable, so you don't actually gain anything by considering sequences indexed by uncountable ordinals