The following is an excerpt from Serge Lang's "Real and Functional Analysis"

In the proof, the author defines $\displaystyle \left \{k...k_n \right \}$ as a subset of $\displaystyle D$.

How does he know that $\displaystyle D$ is big enough to contain a set of elements that can be indexed to $\displaystyle n$?

I assume that, by $\displaystyle n$, the textbook means an index set with the same cardinality as the natural numbers.