3.7 Frequently, eventually[26G]

Let $\mathbb{N}$ be the natural numbers.

This definition is equivalent to definition 176 for real variable $x\to \infty$; it can be further generalized, as seen in 62.

E160

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

Hidden solution: [UNACCESSIBLE UUID ’01B’]

E160

[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’]

E160

[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).

E160

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

• Say what implications there are between

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

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

• Similarly for propositions

  • ”$(P(n)\vee Q(n))$ 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’]

E160

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