Discussion
Consider the set X of all sequences \((x_{n})\) of natural numbers.
Let \(Y = \{(x_{n}) \in X : x_{n} = 0\) for infinitely many \(n\}\),
\(Z = \{(x_{n}) \in X : x_{n} = 0\) for infinitely many \(n\}\),
\(W = \{(x_{n}) \in X : x_{n} \neq 0\) for finitely many \(n\}\).
Then which of the following sets is countable ?
Answer: C
The countable set is W.
W consists of sequences of natural numbers that are non-zero for only finitely many \(n\), that is, sequences that are eventually 0. Each such sequence is determined by a finite list of natural numbers, and the set of finite lists of natural numbers is a countable union of countable sets, hence countable.
The paper prints Y and Z with the same definition (\(x_{n} = 0\) for infinitely many \(n\)), so they are the same set, and it is uncountable: every 0-1 pattern on the odd places with zeros on the even places gives a distinct member, and there are uncountably many such patterns. The complement of Z, sequences with only finitely many zeros, is uncountable too, since it contains all sequences of non-zero natural numbers.
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