4.7 Frequently, eventually[26G]

Let \(ℕ\) be the natural numbers.

Definition 160 frequently, eventually

[018](Solved on 2022-10-27) Let P(n) be a logical clause that depends on a free variable \(n∈ℕ\). We will say that

P(n) holds eventually in \(n\) if

\(∃ m∈ ℕ, ∀ n∈ ℕ\) with \(n≥ m\), P(n) holds ;

P(n) frequently holds in \(n\) if

\(∀ m∈ ℕ, ∃ n∈ ℕ\) with \(n≥ m\) for which P(n) holds.

This definition is equivalent to definition 177 for real variable \(x→∞\); it can be further generalized, as seen in 62.

Remark 161

[23Q]In Italian frequentemente (for frequently) and definitivamente (for eventually) are commonly used in text books; whereas in English these terms are not widely used. 1

E161

[019] Note that «P(n) holds eventually in \(n\)» implies «P(n) holds frequently in \(n\)».

Hidden solution: [UNACCESSIBLE UUID ’01B’]

E161

[01C] Note that «(non P(n)) holds frequently in \(n\)» if and only if «non (P(n) holds eventually in \(n\) )».

Hidden solution: [UNACCESSIBLE UUID ’01D’]

E161

[01F] Note that «P(n) holds frequently in \(n\)» if and only if «P(n) holds for infinitely many \(n\)».

(This equivalence is not true in a generic ordered set. See instead 1 for the correct formulation).

E161

[01G] (Solved on 2022-10-27// in parte) Let now \(P(n),Q(n)\) be two propositions.

  • Say what implications there are between

    • ”\((P(n)∧ Q(n))\) is valid eventually” and

    • ”\(P(n)\) is valid eventually and \(Q(n)\) is valid eventually”.

  • Similarly for propositions

    • ”\((P(n)∨ Q(n))\) is valid eventually” and

    • ”\(P(n)\) is valid eventually or \(Q(n)\) is valid eventually”.

Also formulate similar results for the notion of ”frequently”.

Hidden solution: [UNACCESSIBLE UUID ’01H’]

E161

[29G]Let again \(P(n),Q(n)\) be two propositions. If ”\(P(n)\) is valid eventually and \(Q(n)\) is valid frequently” then ”\((P(n)∧ Q(n))\) is valid frequently”.

  1. With some notable exceptions, such as [ 14 ]