An equivalent condition for infinitely many occurrences

Proved

WorkbookSource.plus_17740

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

We say that $P(n)$ is true infinitely often iff for all $N,$ there exists some $n>N$ such that $P(n)$ is true.

Source: InternLM Lean-Workbook, record lean_workbook_plus_17740 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_17740 (P : ℕ → Prop) : (∀ N, ∃ n > N, P n) ↔ ¬ (∃ N, ∀ n > N, ¬ P n)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_17740; Apache-2.0

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