Question

For every \( n \in \mathbb{N} \), let \( f_n : \mathbb{R} \to \mathbb{R} \) be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]

• A. For every \(x \in \mathbb{R}\) and for every real number \(\varepsilon>0\), there does not exist any integer \(N \ge 0\) such that \(\sum_{i=1}^p |f_{N+i}(x)|<\varepsilon\) for every integer \(p>0.\)
• B. For every \(x \in \mathbb{R}\) and for every real number \(\varepsilon>0\), there exists an integer \(N>0\) such that \(\sum_{i=1}^p |f_{N+i}(x)| \ge \varepsilon\) for some integer \(p>0.\)
• C. There exists \(x \in \mathbb{R}\) and there exists a real number \(\varepsilon>0\) such that for every integer \(N>0\), there exists an integer \(p>0\) for which \(\sum_{i=1}^p |f_{N+i}(x)| \ge \varepsilon.\)
• D. There exists \(x \in \mathbb{R}\) and there exists a real number \(\varepsilon>0\) such that for every integer \(N>0\) and for every integer \(p>0\) the inequality \(\sum_{i=1}^p |f_{N+i}(x)|<\varepsilon\) holds.

Hint:

To negate statements with multiple quantifiers, reverse the order and swap “for all” with “there exists.”

Answer 1

The Correct Option is C

Step 1: Understanding negation.
The original statement uses universal quantifiers (“for every \(x\)”, “for every \(\varepsilon>0\)”...) and existential quantifiers (“there exists \(N\)”). Negation interchanges these quantifiers.
Step 2: Negating logically.
Negation of \[ \forall x \, \forall \varepsilon>0 \, \exists N \, \forall p: P(x, \varepsilon, N, p) \] is \[ \exists x \, \exists \varepsilon>0 \, \forall N \, \exists p: \neg P(x, \varepsilon, N, p). \] That matches option (C).
Step 3: Conclusion.
Hence, option (C) correctly expresses the negation.