Maths WorkshopMaths problems with full solutions
Exercise 1159Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

Quantifiers in convergence of sequences

#convergence#epsilon#sequences

Statement

Write the formal definition of "(un)(u_n) converges to ℓ\ell", then express what it means for (un)(u_n) to be unbounded.

Hint

Convergence: ∀ε>0,∃N,∀n≥N, ∣un−ℓ∣<ε\forall\varepsilon>0,\exists N,\forall n\ge N,\ |u_n-\ell|<\varepsilon. Unbounded: ∀M>0,∃n, ∣un∣>M\forall M>0,\exists n,\ |u_n|>M.

Show solutionHide solution

Solution

Convergence. ∀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.

Unbounded. ∀M>0, ∃n∈N, ∣un∣>M\forall M>0,\ \exists n\in\mathbb{N},\ |u_n|>M.

Conclusion. Both statements use nested quantifiers; unboundedness is a ∀∃\forall\exists statement.

More problems in Quantifiers and negation

Ex.267The existential quantifier
#existential quantifier #exists #example
Quantifiers and negation
High school
High school
Ex.268The universal quantifier
#universal quantifier #forall #proposition
Quantifiers and negation
High school
High school
Ex.269Double negation
#double negation #logic #equivalence
Quantifiers and negation
High school
High school
Ex.270Proving an existential statement
#existential proof #example #witness
Quantifiers and negation
High school
High school