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Eventual domination 2026-10-06

f, g : ω \to ω

, write

f \leq^{*} g

when

f (n) \leq g (n)

for all sufficiently large

n

. A dominating family is cofinal for this relation.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 1 / a / iii / Solution Created 2026-10-03 Updated 2026-10-06

The relation of eventual domination is

f \leq^{*} g ⟺ \exists N \in ω \forall n \geq N f (n) \leq g (n) .

A dominating family

C \subseteq ω^{ω}

satisfies

\forall f \in ω^{ω} \exists g \in C (f \leq^{*} g) .

It must dominate every function, and the exceptional finite initial interval may depend on both functions.

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