of_correctness_atTop_of_terminates

English

If GenContFract.of v terminates, then the convergents stabilize to v at infinity; i.e., for all large enough n, (of v).convs n = v.

Русский

Если GenContFract.of v прерывается, то сходящиеся конвергенты стабилизируются к v на бесконечности; то есть для sufficiently больших n, (of v).convs n = v.

LaTeX

$$$(\\mathrm{Terminates}) \\rightarrow ∀^{\infty} n \\in \\text{atTop},\\ v = (\\mathrm{of}\\, v).convs n$$$

Lean4

/-- If `GenContFract.of v` terminates, then its convergents will eventually always be `v`. -/
theorem of_correctness_atTop_of_terminates (terminates : (of v).Terminates) : ∀ᶠ n in atTop, v = (of v).convs n :=
  by
  rw [eventually_atTop]
  obtain ⟨n, terminatedAt_n⟩ : ∃ n, (of v).TerminatedAt n := terminates
  use n
  intro m m_geq_n
  rw [convs_stable_of_terminated m_geq_n terminatedAt_n]
  exact of_correctness_of_terminatedAt terminatedAt_n

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