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Exercise 329Logic, sets and proofs › Quantifiers and negationL1Added 31 Oct 2026

Negating convergence

#quantifiers#sequences

Statement

Write with quantifiers the statement "the real sequence (un)(u_n) converges to ℓ\ell". Then write its negation, and the negation of "(un)(u_n) converges".

Hint

Swap each ∀\forall with ∃\exists (and the other way round) and negate the final inequality.

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Solution

Convergence to ℓ\ell.
∀ε>0, ∃N∈N, ∀n≥N, ∣un−ℓ∣<ε.\forall\varepsilon>0,\ \exists N\in\mathbb{N},\ \forall n\ge N,\ |u_n-\ell|<\varepsilon.

Negation.
∃ε>0, ∀N∈N, ∃n≥N, ∣un−ℓ∣≥ε.\exists\varepsilon>0,\ \forall N\in\mathbb{N},\ \exists n\ge N,\ |u_n-\ell|\ge\varepsilon.
In words: from some fixed distance ε\varepsilon, infinitely many terms stay at least ε\varepsilon away from ℓ\ell.

Negation of "(un)(u_n) converges". The sequence converges if there exists some ℓ\ell with the property above. Its negation is
∀ℓ∈R, ∃ε>0, ∀N∈N, ∃n≥N, ∣un−ℓ∣≥ε.\forall\ell\in\mathbb{R},\ \exists\varepsilon>0,\ \forall N\in\mathbb{N},\ \exists n\ge N,\ |u_n-\ell|\ge\varepsilon.

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