Denumerability of a subset of a denumerable subset

Sep 2010
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The following is an excerpt from Serge Lang's "Real and Functional Analysis"
Lang.PNG

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.
 
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