A student claims that the family \(\{(0,n): n\in\mathbb{N}, n\ge 1\}\) admits no choice function without the axiom. Which counterexample refutes this claim?
Choose one option.
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C. The function \(f(n)=\tfrac{1}{2}\), which satisfies \(f(n)\in(0,n)\) for all \(n\ge 1\).
Since \(\tfrac{1}{2}\) lies in every interval \((0,n)\) with \(n\ge 1\), the constant function \(\tfrac{1}{2}\) is an explicit choice and the axiom is not needed. Claims about equipotence or countability do not decide by themselves whether a choice function exists.